text:
\section{Introduction}

\subsection{}
Let $\g$ be a simple complex Lie algebra and $\UqLg$ the corresponding (untwisted)
quantum affine algebra. It is well-known that the irreducible finite-dimensional $\UqLg$-representations provide a rich source of rational R-matrices, \ie matrix-valued
rational functions which satisfy the Yang-Baxter equation in a multiplicative parameter.
The goal of the present paper is to show that, relying on the choice of a fixed quantum affine symmetric pair $\Uqk\subset\UqLg$, the same irreducible representations give rise to a family
of rational K-matrices, \ie matrix-valued rational functions which satisfy the reflection equation
(or boundary Yang-Baxter equation) in a multiplicative parameter.\\

The strategy we follow hinges on a recent result of the authors \cite{appel-vlaar-20},
namely the existence of a universal K-matrix for any quantum Kac-Moody algebra.
The latter is a natural operator on category $\cO$ integrable
representations, which depends upon the choice of a quantum symmetric pair and satisfy
a generalized version of the constant (\ie parameter-independent) reflection equation,
featuring a twisting operator $\psi$.
Relying on this construction, we proceed in close similarity to the case of the R-matrix.
We show that the universal K-matrix, modified through the grading shift on $\UqLg$,
can be specialized to a spectral operator $\sKM{V}{z}$ on any finite-dimensional $\UqLg$-representation. Moreover, we prove that
irreducible representations remain irreducible under restriction to the quantum symmetric pair
subalgebra,
thus yielding the existence of a rational operator $\rKM{V}{z}$. Finally, we prove that, for
suitable choices of the twisting operator $\psi$, the generalized reflection equation reduces
to the standard reflection equation. This last step allows us to
recovers many of the known solutions, while providing a large class of entirely new solutions.\\

In the rest of this introduction, we give a brief overview of the main steps of our construction.

\subsection{}
Let $\cV$ be a collection of vector spaces, equipped with two kinds of formal operators. The first, known as \emph{spectral R-matrices}\footnote{We will use the adjective \emph{spectral} instead of \emph{parameter-dependent} to avoid confusion with the dependence on other parameters, notably $q$; the parameter $z$ is known as \emph{spectral parameter} in the literature on integrability.}, are denoted by $\sRM{V \, W}{z}$ ($V,W\in\cV$),
act on tensor products, and satisfy
the \emph{spectral Yang-Baxter equation}:
\eq{
	\label{eq:spectral-YBE-intro}
	\sRM{U \, V}{z} \cdot \sRM{U \, W}{wz} \cdot \sRM{V \, W}{w} = \sRM{V \, W}{w} \cdot \sRM{U \, W}{wz}  \cdot \sRM{U \, V}{z}.
}
The second, known as \emph{spectral K-matrices}, are denoted $\sKM{V}{z}$ ($V\in\cV$),
act on a single vector space, and satisfy, together with the R-matrices, the
\emph{spectral reflection equation} (or \emph{spectral boundary Yang-Baxter equation}):
\eq{
	\label{eq:spectral-RE-intro}
	\begin{aligned}
		& \sRM{W\, V}{w/z}_{21}  \cdot \id\ten\sKM{W}{w} \cdot \sRM{VW}{zw} \cdot \sKM{V}{z} \ten \id = \\
		& \qq \qq = \sKM{V}{z} \ten \id \cdot \sRM{W \, V}{zw}_{21} \cdot \id \ten \sKM{W}{w} \cdot \sRM{VW}{w/z}.
	\end{aligned}
}
where $\sRM{WV}{z}_{21} \coloneqq (1\,2)\circ \sRM{WV}{z} \circ (1\,2)$ and $(1\,2)$ is
the flip operator.

\subsection{}
The spectral reflection equation was introduced
by Cherednik in \cite{cherednik-84} with interlaced mathematical and physical motivations.
\begin{enumerate}\itemsep0.25cm
\item They are consistency conditions for representations of the Artin-Tits braid group of type ${\sf B}_n$ (and, forgetting all equations of the form \eqref{eq:spectral-RE-intro}, representations of the braid group of type ${\sf A}_{n-1}$) on vector-valued formal series of $n$ variables.
Here the group action combines a natural ``tensor'' action on a direct sum of tensor products of the form $V_1 \ot \ldots \ot V_n$ with $V_i \in \mathcal{V}$ and a ``function'' action, through the corresponding Coxeter group $W({\sf B}_n)$ (signed symmetric group, hyperoctahedral group), acting faithfully by permutations and inversions of the variables.
\item They are conditions for factorized particle-particle and particle-boundary scattering on a half-line, involving $n$ particles with state spaces $V_1, \ldots, V_n \in \mathcal{V}$, as a precursor to integrability.
\end{enumerate}
For 1D quantum integrable models (or 2D models from statistical mechanics) with spatial periodicity, the integrability formalism based on the Yang-Baxter equation in terms of transfer matrices and scattering matrices was already well-established in the early 1980s, see e.g. \cite{yang-68,faddeev-79,zamolodchikov-79,baxter-82,gaudin-83,knizhnik-84}.
For 1D models with two boundaries,
the formalism was developed in \cite{sklyanin-88,cherednik-91,cherednik-92}
requiring as additional input solutions of \eqref{eq:spectral-RE-intro} and of a similar equation associated to the other boundary (see \cite{vlaar-15} for the connection between two-boundary transfer matrices and scattering matrices).

\subsection{}
It is natural to consider the case that $\mathcal{V}$ is a suitable category of modules of a (unital associative) algebra $A$.
Namely, in this case it is conceivable that the family of equations \eqref{eq:spectral-YBE-intro} can be obtained by applying suitable representation maps to a single equation in $A \otimes A \otimes A$, or rather its completion with respect to the category $\mathcal{V}$.
A similar approach is a priori sensible for \eqref{eq:spectral-RE-intro}.\\

Let $q$ be a formal parameter, $\bsF\coloneqq\ol{\bbC(q)}$, and $\mathcal{V}$ the category of finite-dimensional representations\footnote{We only consider so-called type $\bf 1$ representations of quantum groups.} of the (untwisted) quantum affine algebra $U_qL\fkg$ associated to the
loop algebra $L\fkg \cong \fkg \otimes \C[z,z^{-1}]$ of a finite-dimensional simple Lie algebra $\fkg$.
The affine quantum group $U_q\wt\fkg$, like any quantized Kac-Moody algebra, is a quasitriangular Hopf algebra with respect to its category $\cO$; that is, in the completion of the tensorial square of $U_q\wt\fkg$ with respect to $\cO$ lies an element $R$, called \emph{universal R-matrix}, which satisfies the \emph{universal Yang-Baxter equation}
\eq{ \label{eq:universal-YBE-intro}
R_{12} \cdot R_{13} \cdot R_{23} = R_{23} \cdot R_{13} \cdot R_{12}.
}
Drinfeld \cite{drinfeld-quantum-groups-87} used properties of $R$ to show the existence of a power series $R(z)$ taking values in a completion of the tensorial square of $U_qL\fkg$ with respect to its category of finite-dimensional modules.
This $R(z)$ satisfies a universal version of \eqref{eq:spectral-YBE-intro}, i.e. a spectral version of \eqref{eq:universal-YBE-intro}.
By setting $\sRM{V \, W}{z}$ equal to the grading-shifted action of this power series on $V \ot W$ for any $V,W \in \mathcal{V}$, the existence of solutions of \eqref{eq:spectral-YBE-intro} for any $U, V, W \in \mathcal{V}$ readily follows.
Moreover, if $V,W \in \mathcal{V}$ are irreducible $U_qL\fkg$-modules then it is known that the tensor product $V \ot W$ is \emph{generically irreducible}, i.e. the formal Laurent series extension $V \ot W((z))$ is irreducible as a module of $U_qL\fkg \ot \bsF((z))$, and hence $\sRM{V \, W}{z}$, up to a scalar-valued formal power series, depends rationally on $z$, see e.g. \cite{frenkel-reshetikhin-92,kazhdan-soibelman-95}.

\subsection{}
In this paper we consider an analogous universal approach to the construction of solutions of the spectral reflection equation.
It is useful to remark here on a crucial subtlety compared to the spectral Yang-Baxter equation.
Namely, one may in first instance consider a reflection equation of the form
\eq{ \label{eq:universal-REnaive-intro}
R_{21} \cdot 1 \ot K \cdot R \cdot K \ot 1 = K \ot 1 \cdot R_{21} \cdot 1 \ot K \cdot R
}
where $K$ lies in (a suitable completion of) a quantum group $U_q\fkg$ of finite type, see \cite{dieck-98,balagovic-kolb-19} for bialgebraic and categorical-topological formalisms, as well as explicit constructions.
This yields solutions of the \emph{constant} (i.e. parameter-independent) reflection equation in finite-dimensional modules of $U_q\fkg$.
However, proposing this as the universal version of \eqref{eq:spectral-RE-intro} in $U_qL\fkg$ will not do: it is easy to see that applying grading-shifted actions on $V,W \in \mathcal{V}$ instead yields
\eq{
\label{eq:bad-spectral-RE-intro}
\begin{aligned}
& \sRM{W\, V}{z/w}_{21}  \cdot \id\ten\sKM{W}{w} \cdot \sRM{VW}{w/z} \cdot \sKM{V}{z} \ten \id = \\
& \qq \qq = \sKM{V}{z} \ten \id \cdot \sRM{W \, V}{w/z}_{21} \cdot \id \ten \sKM{W}{w} \cdot \sRM{VW}{w/z}.
\end{aligned}
}
This equation corresponds instead to a \emph{non-faithful} action of the Coxeter group $W({\sf B}_n)$ on functions or formal series on $n$ variables.
More precisely, if we define $W({\sf B}_n)$ as usual with generators $s_0,s_1,\ldots,s_{n-1}$ and relations
\eq{
\begin{aligned}
s_i s_{i+1} s_i &= s_{i+1} s_i s_{i+1} && \text{if } 0<i<n-1, \qq \qq \qq &
s_0s_1s_0s_1 &= s_1s_0s_1s_0, \\
s_is_j &=s_js_i && \text{if } j>i+1, &
s_i^2 &= 1,
\end{aligned}
}
then here we are referring to the action which trivially extends the permutation action of the symmetric group: each $s_i$ ($0<i<n$) acts as a simple transposition on the variables and $s_0$ acts trivially.

\subsection{}
It is instead natural to consider a \emph{twisted universal reflection equation}, namely an equation of the form
\eq{ \label{eq:universal-REtwisted-intro}
(R^{\psi \psi})_{21} \cdot 1 \ot K \cdot R^\psi \cdot K \ot 1 = K \ot 1 \cdot (R^\psi)_{21} \cdot 1 \ot K \cdot R,
}
where $\psi$ is an algebra automorphism of $U_qL\fkg$ and $R^\psi:=(\psi \ot \id)(R)$, $R^{\psi \psi} := (\psi \ot \psi)(R)$.
Such universal equations and the supporting bialgebraic and categorical frameworks were considered in \cite{appel-vlaar-20}, with a concrete realization in the context of representation categories $\cO$ and $\cO^{\sf int}$ of q-deformed enveloping algebras of symmetrizable Kac-Moody algebras.
In the present paper we restrict to the affine case and develop this formalism further in the context of finite-dimensional representations of $U_qL\fkg$.

Note that, in analogy with R-matrices, the universal K-matrix has an intertwining property.
However, in order to state this we need additional datum in the form of a quantum pseudo-symmetric pair $(U_q\wt\fkg,U_q\fkk)$, see \cite{kolb-14,regelskis-vlaar-21}, where $U_q\fkk$ is a right coideal subalgebra of $U_q\wt\fkg$ mimicking the fixed-point subalgebra of an involution of $U_q\wt\fkg$ of the \emph{second kind}, cf. \cite[4.6]{kac-wang-92}.
In this paper we will call such algebras $U_q\fkk$ \emph{(affine) QSP subalgebras}.
For a given affine QSP subalgebra $U_q\fkk$, the element $K$ enjoys a universal intertwining property with respect to $U_q\fkk$: it lies in the $\psi$-twisted centralizer of $U_q\fkk$ (in fact a modified version of this statement is used to define $K$).

\subsection{}
One of the main results of this paper is that, for all $V \in \mathcal{V}$ and all affine QSP subalgebras $U_q\fkk$, there exist twists $\psi$ such that the universal element $K$ evaluates to an $\End(V)$-valued formal Laurent series $K_V(z)$, and the intertwining property and the universal twisted equation \eqref{eq:universal-REtwisted-intro} descend to equations for $K_V(z)$.
Namely, if $\pi_{V,z}$ is the grading-shifted representation map for $V$ and $\psi^*(V)$ is the pullback of $V$ (i.e. the module where the algebra action has been twisted by $\psi$), then we have
\eq{ \label{eq:twisted-intw-intro}
K_V(z) \pi_{V,z}(b) = \pi_{\psi^*(V),1/z}(b) K_V(z) \qq \text{for all } b \in U_q\fkk,
}
and, for $V,W \in \mathcal{V}$,
\eq{
\begin{aligned}
\label{eq:generalized-spectral-RE-intro}
	\sRM{\psi^*(W)\, \psi^*(V)}{w/z}_{21}&\cdot \id\ten\sKM{W}{w}
	\cdot\sRM{\psi^*(V)W}{zw}\cdot\sKM{V}{z}\ten \id=\\
	&=\sKM{V}{z}\ten \id\cdot\sRM{\psi^*(W)\, V}{zw}_{21}\cdot \id\ten\sKM{W}{w}
	\cdot\sRM{VW}{w/z}.
\end{aligned}
}
Precisely this generalized reflection equation was considered in \cite{cherednik-92} and used to set up a generalized formalism quantum Knizhnik-Zamolodchikov equations; also cf. \cite{freidel-maillet-91} for similar generalized reflection equations.
It turns out that, to obtain the spectral inversion associated to the ``function action'' of the generator $s_0 \in W({\sf B}_n)$ in \eqref{eq:twisted-intw-intro} and \eqref{eq:generalized-spectral-RE-intro}, it is essential that $\psi$ deforms an automorphism of $\wt \fkg$ of the second kind.

\subsection{}
Secondly, we show that, if $V \in \mathcal{V}$ is an irreducible $U_qL\fkg$-module then it remains so as a module of the q-deformed standard nilpotent subalgebra, building on results by \cite{chari-greenstein-05, bowman-07, hernandez-jimbo-12} for the corresponding q-deformed Borel subalgebra.
We then promote this to \emph{generic QSP irreducibility}, i.e. $V \otimes \bsF((z))$ is irreducible as a $U_q\fkk \otimes \bsF((z))$-module, provided the grading shift is the principal one.
Its main consequence is that the solution space of \eqref{eq:twisted-intw-intro} is one-dimensional (for any suitable grading shift).
In analogy with R-matrix theory, we deduce that $K_V(z)$ factorizes as a product of a scalar-valued formal Laurent series and an $\End(V)$-valued rational function, which itself satisfies the generalized reflection equation \eqref{eq:generalized-spectral-RE-intro}.
This connects with the literature in quantum integrability on the classification and intertwining properties of trigonometric\footnote{
Regarding the nomenclature, observe that rational functions in the (multiplicative) parameter $z$ can naturally be rewritten as functions depending trigonometrically on an (additive) parameter proportional to $\log(z)$.
}
K-matrices, see e.g. \cite{delius-02,delius-george-02,delius-mackay-03,malara-lima-santos-06,regelskis-vlaar-16,regelskis-vlaar-18}.

\subsection{Outline}
In Section~\ref{s:affine}, we recall the definition of quantum affine algebras and the constrution of the rational R-matrix on irreducible finite-dimensional representations.
In Section~\ref{s:affine-qsp}, we recall the definition of QSP subalgebras and provide a brief review of the construction of universal K-matrices for quantum Kac-Moody algebras as given in \cite{appel-vlaar-20}.
In Section~\ref{s:spectral}, we present the first main result of the paper. Namely,
the construction of parameter-dependent K-matrices on finite-dimensional representations is described in Theorem~\ref{thm:spectral-k}. In Section~\ref{s:rational}, we prove in Theorem~\ref{thm:spectral-k} that this yields the existence of rational K-matrices for irreducible representations. Moreover, we discuss unitarity and normalization under additional assumptions in \ref{ss:unitarity}-\ref{ss:abelian-k-mx}.
In Section~\ref{s:irreducibility}, we prove the second main result of the paper, which Theorems~\ref{thm:rational-k}, \ref{thm:spectral-k} rely on.
Namely, we prove in Theorem~\ref{thm:deformedN:irreducible} that every irreducible representation over $\UqLg$ remains generically irreducible under restriction to a large class of subalgebras, including all QSP subalgebras.
Finally, in Section~\ref{s:recover-RE}, we show that our construction recovers many of the known solutions of the standard reflection equation and most importantly gives rise to a
large class of entirely new solutions.

\subsection{Acknowledgments}
The authors would like to thank Martina Balagovi\'{c}, Vijayanthi Chari, Gustav Delius, Sachin Gautam, Tomasz Prze\'{z}dziecki, Catharina Stroppel, Valerio Toledano Laredo, and Curtis Wendlandt for their interest in this work and for useful discussions.
The essence of this paper was conceived during the hybrid Mini-Workshop \emph{Three Facets of R-Matrices} at the Mathematisches Forschungsinstitut Oberwolfach.
We are grateful to the organizers for their invitation and to the institute for the wonderful working conditions.

\section{Quantum affine algebras}\label{s:affine}

In this section we recall the definition of quantum affine algebras and basic results on
their irreducible finite-dimensional representations. Given a lattice $\mathsf{\Lambda}$, we shall denote the non-negative part by $\mathsf{\Lambda}_+$.

\subsection{Untwisted affine Lie algebras}\label{ss:affine}

Let $\g$ be a finite-dimensional simple Lie algebra defined over\footnote{
The field can be any algebraically closed field of characteristic zero.
For simplicity, we shall work with $\bbC$ in order to emphasize the analytic properties of R-matrices and K-matrices.
} $\bbC$ with Cartan subalgebra $\h\subset\g$.
Let $\fIS\coloneqq\{1,2,\dots, \rank(\g)\}$ be the set of vertices of the corresponding Dynkin diagram, $A=(a_{ij})_{i,j\in\fIS}$ the Cartan matrix, $\iip{\cdot}{\cdot}$ the normalized invariant bilinear form on $\g$, $\Pi\coloneqq\{\rt{i}\;|\; i\in\fIS\}\subset\h^*$ a basis of simple roots and $\Pi^{\vee}\coloneqq\{\cort{i}\;|\; i\in\fIS\}\subset\h$ a basis of simple coroots such that $\cp{\rt{j}}{\cort{i}}=a_{ij}$ for all $i,j \in I$.
Let $\Qlat\coloneqq\bbZ\Pi\subset\h^*$ and $\Qlat^{\vee}\coloneqq\bbZ\Pi^\vee\subset\h$ be the root and coroot lattice, respectively.
Let $\rootsys_+\subset\Qp$ be the set of positive roots
and $\hrt=\sum_{i\in\fIS}a_i\rt{i}$ the highest root.
Finally, let $\Plat\coloneqq\{\lambda\in\h^*\;|\;\lambda(\Qlatv)\subset\bbZ\}$ be the weight lattice.\\

Let ${\ag}$ be the (untwisted) affine Lie algebra associated to $\g$
with affine Cartan subalgebra ${\ah}\subset{\ag}$ \cite[Ch.~7]{kac-90}.
Set $\agp\coloneqq\ag^\prime$ and $\ahp\coloneqq\ah^\prime$.
Let $\aIS\coloneqq\{0\}\cup\fIS$ be the set of vertices of the affine Dynkin diagram
and $\aff{A}=(a_{ij})_{i,j\in\aIS}$ the extended Cartan matrix \cite[Table Aff. 1]{kac-90}.
We denote by $\aQv \subset {\ah}$ and $\aQ \subset {\ah}^*$ the affine coroot and root lattices, respectively, and set ${\ahp} := \C \aQv \subseteq {\ah}$ and $({\ah}^*)' := \C \aQ \subseteq {\ah}^*$.
Let $\drv{}\in\aQp$ and $c\in\aQvp$ be the unique elements such that
\begin{align}
	\{\lambda\in\aQ\;|\; \forall i\in\aIS,\, \cp{\lambda}{\cort{i}}=0 \}=\Z\drv{}\aand
	\{h\in\aQv\;|\; \forall i\in\aIS,\, \cp{\rt{i}}{h}=0 \}=\Z \central{}
\end{align}
In particular, $\drv{}=\rt{0}+\hrt$, $c$ is central in ${\ag}$, and, under the identification $\nu:{\ah}\to{\ah}^*$ induced by the bilinear form, one has $\nu(c)=\delta$.
The sets of real and imaginary affine positive roots in $\aQp$ are described by
\begin{equation}
\aff{\Delta}_+^{\sf re}=\Delta_++\bbZ_{\geqslant0}\drv{}
\aand
\aff{\Delta}_+^{\sf im}=\bbZ_{>0}\drv{}\, .
\end{equation}
We fix $\codrv{}\in{\ah}$ such that $\cp{\rt{i}}{\codrv{}}=\drc{i0}$ for any $i\in\aIS$.
Note that $\codrv{}$ is defined up to a summand proportional to $c$ and we obtain a natural identification ${\ah}=\h\oplus\bbC c\oplus\bbC\codrv{}$.
In terms of the \emph{extended} coroot lattice $\aQvext\coloneqq\aQv\oplus\bbZ d\subset{\ah}$ we set $\aP\coloneqq\{\lambda\in{\ah}^*\;|\;\lambda(\aQvext)\subset\bbZ\}$.
Then, the quotient lattice\footnote{$\aPd$ is also referred to as the {\em classical} weight lattice (see \eg \cite{kashiwara-02}).}
\[
\aPd\coloneqq\aP/(\aP\cap\bbQ\drv{})\simeq\hom_{\bbZ}(\aQv,\bbZ)
\]
has a basis given by the images of the fundamental weights in $\aP$.

\subsection{Drinfeld-Jimbo presentation of the affine quantum group} \label{ss:quantumaffine}

Let $q$ be an indeterminate, let $\bsF$ the field of rational functions in $q$ with complex coefficients.
Consider the algebraic closure $\bsF=\ol{\bbC(q)}$.
Fix non--negative integers $\{\de{i}\;|\; i\in\aIS\}$ such that the matrix $(\de{i}a_{ij})_{i,j\in\aIS}$ is symmetric and set $q_i\coloneqq q^{\de{i}}$.
The quantum Kac--Moody algebra associated to ${\ag}$ is the algebra $\Uqag$ over $\bsF$ with generators $\Eg{i},\Fg{i}$, $i\in\aIS$, and $\Kg{h}$, $h\in\aQvext$ subject to the following defining relations:
\begin{gather}
	\Kg{h}\Kg{h'}=\Kg{h+h'}, \qq \Kg{0}=1, \\
	\Kg{h}\Eg{i}=q^{\cp{\rt{i}}{h}}\Eg{i}\Kg{h}, \qq \Kg{h}\Fg{i}=q^{-\cp{\rt{i}}{h}}\Fg{i}\Kg{h}, \\
	[\Eg{i},\Fg{j}]=\drc{ij}\frac{\Kg{i}-\Kg{i}^{-1}}{q_i-q_i^{-1}}, \label{Uqag:relns3} \\
	\mathsf{Serre}_{ij}(\Eg{i},\Eg{j})=0=\mathsf{Serre}_{ij}(\Fg{i},\Fg{j})\quad(i\neq j) \label{Uqag:relns4}
\end{gather}
for any $i,j\in\aIS$ and $h,h'\in\aQvext$, where $\Kg{i}^{\pm1}\coloneqq\Kg{\pm \de{i}\cort{i}}$
and the last line contains to the usual q-deformed Serre relations from e.g. \cite[3.1.1 (e)]{lusztig-94}.

We denote by $\Uqg$ the subalgebra generated by $\Eg{i}, \Fg{i}$, $i\in\fIS$, and $\Kg{h}$, $h\in\Qlatv$, which is a quantum Kac-Moody algebra of finite type.

\begin{remark}
		For any $\mu\in\aQ$, we set $\Kg{\mu}\coloneqq\Kg{\nu^{-1}(\mu)}$, where
	$\nu^{-1}:\aQ\to\aQv$ is induced by the identification $\nu:{\ah}\to{\ah}^*$.
	In particular, $\Kg{i}^{\pm1}=\Kg{\pm\rt{i}}$.
\end{remark}

On $\Uqag$ we shall consider the Hopf algebra structure determined by
\begin{align}
\Delta(\Kg{h}) &= \Kg{h}\ten\Kg{h}, & 	\Delta(\Eg{i}) &= \Eg{i}\ten1+\Kg{i}\ten\Eg{i}, &	\Delta(\Fg{i}) &= \Fg{i}\ten\Kg{i}^{-1}+1\ten\Fg{i},
\end{align}
for any $i\in\aIS$ and $h\in\aQvext$.
Finally, the Chevalley involution $\chev:\Uqag\to(\Uqag)^{\scsop{cop}}$ is the isomorphism of Hopf algebras defined by
\begin{equation}\label{eq:chevalley}
	\chev(\Kg{h})=\Kg{-h},\quad \chev(\Eg{i})=-\Fg{i},\quad \chev(\Fg{i})=-\Eg{i}\,.
\end{equation}
for any $i\in\aIS$ and $h\in\aQvext$.\\

We denote by $\Uqanp$ and $\Uqanm$ the subalgebras generated by the elements $\{\Eg{i}\}_{i\in\aIS}$ and $\{\Fg{i}\}_{i\in\aIS}$, respectively.
We set $\Uqabpm\coloneqq\Uqanpm\Uqah$, where $\Uqah$ is the commutative subalgebra generated by $\Kg{h}$, $h\in\aQvext$.
Similarly, we denote by $\Uqh, \Uqnpm, \Uqbpm\subset\Uqg$ their {finite-type} counterparts.

Finally, we denote by $\Uqagp$ the subalgebra obtained by replacing the extended coroot lattice $\aQvext$ with the standard coroot lattice $\aQv$; note that it is generated by $E_i$, $F_i$, $K_i^\pm$ for all $i \in \wh I$.\\

Consider the central element $\Kg{c}=\Kg{\drv{}}=\Kg{0}\Kg{\hrt}=\Kg{0}\prod_{i\in\fIS}\Kg{i}^{a_i}$.
The \emph{quantum loop algebra} $\UqLg$ is the quotient of $\Uqagp$ by the ideal generated by $\Kg{c}-1$.
Note that the Hopf algebra structure and the Chevalley involution descend to $\UqLg$.

\subsection{Category $\cO$ and finite--dimensional representations}\label{ss:QL-representations}
In the case of $\Uqg$, the irreducible objects in the categories of finite--dimensional (type $\mathbf{1}$) representations and integrable category $\cO$ representations coincide.
However, for quantum affine algebras, the two categories share only trivial representations and have significantly different features, as we briefly describe below.\\

Let $V\in\Rep(\Uqag)$.
Recall that $V$ is
\begin{enumerate}\itemsep0.25cm
	\item a (type $\mathbf{1}$) weight representation if $V=\bigoplus_{\mu\in\aP}\wsp{V}{\mu}$, where
	\begin{equation}
	\wsp{V}{\mu}\coloneqq\{v\in V\;|\; \forall h\in\aQvext,\, \Kg{h}\cdot v=q^{\cp{\mu}{h}}v\}\, ;
	\end{equation}

	\item an integrable representation if it is a weight representation and the action of the elements $\{\Eg{i}, \Fg{i}\;|\; i\in\aIS\}$ is locally nilpotent;

	\item a category $\cO^{\pm}$ representation if it is a weight representation and the action of
	$\Uqanpm$ is locally finite\footnote{In the case $\cO^+$, this coincides with the category $\cC^{\scsop{hi}}$ from \cite[Sec.~3.4.7]{lusztig-94}. Note that $V$ is not required to be finitely-generated nor to have finite-dimensional weight spaces.} (for a choice of sign).

\end{enumerate}

We denote by $\Ointpm(\Uqag)$ the full subcategory of integrable category $\cO^{\pm}$ representations.
It is well--known that $\Ointpm(\Uqag)$ is semisimple and that the nontrivial irreducible representations in $\Ointpm(\Uqag)$ are infinite--dimensional and classified by non--zero dominant weights, see \cite[Thm~6.2.2, Cor.~6.2.3]{lusztig-94}.
Moreover, $\Ointpm(\Uqag)$ has a natural tensor product and a braiding (cf.~\ref{ss:R-matrix}).\\

Let $V\in\Repfd(\Uqagp)$. Recall that $V$ is a {type $\mathbf{1}$} representation if the generators $\Kg{i}$ ($i\in\aIS$) act semisimply on $V$ with eigenvalues in $q^{\bbZ}$ and the central element $\Kg{c}$ acts as $1$ on $V$.
Note that $V$ admits a weight decomposition over the quotient lattice $\aPd$.

By \cite[Prop.~12.2.3]{chari-pressley}, every irreducible representation in $\Repfd(\Uqagp)$ is twist-equivalent to a type $\mathbf{1}$ representation.
In particular, the problem of studying the finite--dimensional representation theory of $\Uqagp$ essentially reduces to studying that of the quantum loop algebra $\UqLg$.
This is known to be extremely rich.
First of all, $\Repfd(\UqLg)$ is not semisimple; nevertheless its irreducible representations are classified, namely by $\rank(\g)$--tuples of monic polynomials over $\bsF$, see \cite[Thm.~12.2.6]{chari-pressley}.
Moreover, although $\Repfd(\UqLg)$ is monoidal, it is not braided in the usual sense (cf.~Section \ref{ss:spectral-R}).

\subsection{Completions}\label{ss:completions}
In the following, we shall consider suitable completions of the algebras $\Uqag$  and $\UqLg$, so as to include certain distinguished operators acting on representations in $\Ointpm(\Uqag)$ and $\Repfd(\UqLg)$, respectively.\\

Let $A$ be a bialgebra and $\cC\subseteq\Rep(A)$ a monoidal subcategory.
We denote by $A^{\cC}$ the algebra of endomorphisms of the forgetful functor $\cC\to\Vect{}$.
Recall that an element of $A^{\cC}$ is a collection of linear maps $\{\cE_{V}\}_{V\in\cC}$ such that, for any intertwiner $f:V\to W$, one has $f\circ\cE_V=\cE_W\circ f$.
Note that there is a natural algebra map $\iota: A\to A^{\cC}$ and $\cC$ is said to \emph{separate points} if $\iota$ is an embedding.
Similarly, for any $n>0$, we denote by $(A^{\ten n})^{\cC}$ the algebra of endomorphisms of the $n$-fold forgetful functor $\cC^{n}\to\Vect{}$, given by $(V_1,\dots, V_n)\mapsto V_1\ten\cdots\ten V_n$.

The monoidal structure on $\cC$ induces on the tower of algebras $(A^{\ten n})^{\cC}$ ($n\geqslant1$) the structure of a \emph{cosimplicial algebra} (see \eg \cite[Sec.~2.10-2.11]{appel-vlaar-20}, \cite[Sec.~8.9]{appel-toledano-19b}). Roughly, this means that $A^{\cC}$ can be thought of as
a topological bialgebra, whose structure extends that of $A$ through $\iota$.
Similarly, every automorphism $\phi:A\to A$, whose pullback preserves
$\cC$, naturally extends to $A^{\cC}$ by $\phi(\cE_V)=\cE_{\phi^*(V)}$.\\

We shall be interested in the algebras $\CUqag{}$ and $\CUqLg{}$, obtained as the completion of $\Uqag$ and $\UqLg$ with respect to categories $\Ointp(\Uqag)$ and $\Repfd(\UqLg)$, respectively.

\subsection{The universal R-matrix}\label{ss:R-matrix}

The Hopf algebra $\Uqag$ is {quasitriangular}, \ie it admits a universal R-matrix $\RM{}\in\Uqabm\widehat{\ten}\Uqabp$, where $\wh{\ten}$ denotes the completion with respect to the $\Qlat$--grading, satisfying the intertwining equation $\RM{}\Delta(x)=\Deltaop(x) R$ for any $x\in\Uqag$ and the coproduct identities
\begin{equation}
\Delta\ten\id(\RM{})=\RM{13} \RM{23}
\aand
\id\ten\Delta(\RM{})=\RM{13} \RM{12}\, .
\end{equation}
In particular, it follows that $\RM{}$ satisfies the Yang--Baxter equation
\eq{
\RM{12} \RM{13} \RM{23} = \RM{23} \RM{13} \RM{12}.
}
The operator $\RM{}$ arises from the Drinfeld double construction of $\Uqag$ as the canonical tensor of a Hopf pairing between $\Uqabm$ and $\Uqabp$
(cf \cite{drinfeld-quantum-groups-87} and \cite[Ch.~4]{lusztig-94}).\\

More specifically, let $\{u_i\}, \{u^i\}\subset\h$ be dual bases and set
\eq{
\Omega_0\coloneqq\sum_iu_i\ten u^i, \qq \wh{\Omega}_0\coloneqq m(c\ten d+d\ten c)+\Omega_0,
}
where $m=1,2,3$ if $\g$ is of type $\mathsf{ADE}$, $\mathsf{BCF}$, or $\mathsf{G}$, respectively.
Then the R-matrix of $\Uqag$ has the form
\begin{align}\label{eq:R-mx}
	\RM{}=q^{\wh{\Omega}_0}\cdot\sum_{\mu>0}\Xi_{\mu}\in\Uqabm\wh{\ten}\Uqabp\ ,
\end{align}
where $\Xi_{\mu}\in\Uqanm_{-\mu}\ten\Uqanp_{\mu}$ is the $\mu$--component of the canonical tensor\footnote{Note that the operator $q^{\wh{\Omega}_0}$ is denoted $\ka_{\sf id}$ in \cite{appel-vlaar-20, balagovic-kolb-19}). See also Remark~\ref{rmk:R-action}.}.
Finally, note that $\omega\ten\omega(\RM{})=\RM{21}$.
Hence the Chevalley involution is an isomorphism of the quasitriangular Hopf algebras $U_q\ag$ and $U_q\ag^{\sf cop}$.

\begin{remark}\label{rmk:R-action}
The universal R-matrix is properly defined as an element of the completion $(\Uqag\ten\Uqag)^{\cO^{\pm}}$ with respect to category $\cO^{\pm}$ representations.
Indeed, $R$ does not lie in $\Uqag\ten\Uqag$, but it does evaluate to a well-defined operator $R_{VW}$ on any tensor product $V\ten W$ of category $\cO^{\pm}$ representations.
Note, in particular, that the action of  $q^{\wh{\Omega}_0}$ on the tensor product of weight vectors is given by $q^{\wh{\Omega}_0}\cdot v\ten w = q^{\iip{\wgt{v}}{\wgt{w}}}\cdot v\ten w$.
Clearly, $R$ defines a braiding in the category $\cO^{\pm}(\Uqag)$.
\rmkend
\end{remark}

\subsection{The spectral R-matrix}\label{ss:spectral-R}
The universal R-matrix of $\Uqag$ does not immediately act on finite--dimensional
representations $\UqLg$. The first obstacle is given by the operator
$q^{m(c\ten d+d\ten c)}$. However, this is easily solved by observing that, since the central
element $\Kg{c}$ acts by $1$ if and only if $c$ acts by zero, that factor can be ignored.
The second obstacle is given by the fact that the projection of the operator
$\Xi\coloneqq\sum_{\mu>0}\Xi_{\mu}$ does not necessarily converge on finite--dimensional
representations over $\UqLg$.
To this end, set
\eq{
\Uqag[z,z^{-1}]\coloneqq\Uqag\ten\bbF[z,z^{-1}]
}
and consider the \emph{homogeneous grading shift} automorphism
\begin{equation}\label{eq:grading-shift}
	\shift{z}: \Uqag[z,z^{-1}]\to\Uqag[z,z^{-1}]
\end{equation}
given by $\shift{z}(\Kg{h})\coloneqq\Kg{h}$, $\shift{z}(\Eg{i})\coloneqq z^{\drc{i0}}\Eg{i}$,
and $\shift{z}(\Fg{i})\coloneqq z^{-\drc{i0}}\Fg{i}$. Note that, by specializing $z$
in $\bsF$, we obtain a one--parameter family of automorphism of $\Uqag$.
Then, let
\begin{equation}\label{eq:shift-coproduct}
	\Delta_{z}, \Deltaop_{z}:\Uqag[z,z^{-1}]\to(\Uqag\ten\Uqag)[z,z^{-1}]
\end{equation}
be the \emph{shifted coproducts} defined by
\eq{
\Delta_z(x)\coloneqq\id\ten\shift{z}(\Delta(x)), \qq \Deltaop_z(x)\coloneqq\id\ten\shift{z}(\Deltaop(x)).
}

The grading shift is clearly well--defined on $\UqLg$.
For any $V\in\Repfd(\UqLg)$ with action $\pi_V:\UqLg\to\End(V)$, we consider the infinite--dimensional representations
\eq{
\shrep{V}{z}\coloneqq V\ten\bsF(z), \qq \Lshrep{V}{z}\coloneqq V\ten\Lfml{\bsF}{z},
}
with action given by $\pi_V(\shift{z}(x))$. By considering the projection of the formal series $\id\ten\shift{z}(\RM{})\in\fml{(\Uqag\wh{\ten}\Uqag)}{z}$ on the quantum loop algebra,
one obtains the following theorem due to Drinfeld (cf.~\cite{drinfeld-quantum-groups-87}, see
also \cite{frenkel-reshetikhin-92, hernandez-19}).

\begin{theorem}\label{thm:spectral-R}\hfill
\begin{enumerate}\itemsep0.25cm
\item The quantum loop algebra $\UqLg$ has a universal \emph{spectral} R-matrix, \ie
a distinguished element $\sRM{}{z}\in\fml{(\UqLg\wh{\ten}\UqLg)}{z}$ such that
$\shift{a}\ten\shift{b}(\sRM{}{z})=\sRM{}{\frac{b}{a}z}$ ($a,b\in\bsF^{\times}$) and the
following identities are satisfied:
\begin{gather}
	\sRM{}{z}\Delta_{z}(x) = \Deltaop_{z}(x)\sRM{}{z} \qu \text{for all } x \in \UqLg ,\\
	\Delta_z\ten\id(\sRM{}{zw}) = \sRM{13}{zw} \sRM{23}{w}, \qq \id\ten\Delta_w(\sRM{}{z}) = \sRM{13}{z} \sRM{12}{zw}\, .
\end{gather}
In particular, the spectral Yang--Baxter equation holds:
\begin{equation}\label{eq:z-YBE}
\sRM{12}{z} \sRM{13}{zw} \sRM{23}{w} = \sRM{23}{w} \sRM{13}{zw} \sRM{12}{z}\, .
\end{equation}
\item For any $V,W\in\Repfd(\UqLg)$, the operator
\begin{equation}\label{eq:sRM-rep}
	\sRM{VW}{z}\coloneqq\pi_{V}\ten\pi_W(\sRM{}{z})
	\in\fml{\End(V\ten W)}{z}
\end{equation}
is well--defined and provides an intertwiner
 \begin{equation}\label{eq:sRM-rep}
 	\sRMv{VW}{z}\coloneqq(1\,2)\circ\sRM{VW}{z}:\Lshrep{V}{z}\ten W\to W\ten\Lshrep{V}{z}\, .
 \end{equation}
\end{enumerate}
\end{theorem}

\subsection{Rational R-matrices}\label{ss:rational-R}
In the case of irreducible representations the operator $\sRM{VW}{z}$ features the
following rationality property (see \eg~\cite{jimbo-86, drinfeld-87, frenkel-reshetikhin-92, kazhdan-soibelman-95,etingof-frenkel-kirillov-98}).

\begin{theorem}\label{thm:rational-R}
Let $V,W\in\Repfd(\UqLg)$ be two irreducible representations.
There exists a canonical scalar-valued formal Laurent series $f_{VW}(z)\in\Lfml{\bsF}{z}$ such that
\begin{equation}
\rRM{VW}{z}\coloneqq f_{VW}(z)^{-1}\sRM{}{z}\in\Lfml{\End(V\ten W)}{z}
\end{equation}
is rational and satisfies the spectral Yang-Baxter equation \eqref{eq:z-YBE} and the unitarity relation
\eq{ \label{eq:R:unitarity}
\rRM{VW}{z}^{-1}=(1\, 2)\circ\rRM{WV}{z^{-1}}\circ(1\, 2).
}
In particular, $\rRMv{VW}{z}\coloneqq(1\,2)\circ\rRM{VW}{z}$
is an intertwiner $V\ten\shrep{W}{z}\to \shrep{W}{z}\ten V$.
\end{theorem}

The proof of the theorem relies on the \emph{generic irreducibility} of the tensor product
$V\ten W$, \ie on the irreducibility of the representation $V\ten\Lfml{W}{z}$ over $\Lfml{\UqLg}{z}$ (cf.~ \cite[Sec.~4.2]{kazhdan-soibelman-95} or \cite[Thm.~3]{chari-02}).
We shall use a similar argument in Section~\ref{s:rational} to prove the existence of a rational K-matrix.
Finally, note that $f_{VW}(z)$ is uniquely determined by the condition $\rRM{}{z}(v\ten w)=v\ten w$, where $v\in V$ and $w\in W$ are highest weight vectors.

\begin{example}
Let $\g=\mathfrak{sl}(2)$, $V_1=\bbC^2$ the fundamental representation, and
$\evrep{1}{a}$ the corresponding evaluation representation of $U_qL\mathfrak{sl}(2)$ at
$a\in\bbC^\times$. In the case of $\evrep{1}{a}\ten\evrep{1}{bz}$, the rational function $\rRM{a,b}{z}\coloneqq\rRM{\evrep{1}{a}\,\evrep{1}{b}}{z}$ is easily computed (see \eg
\cite[12.5.7]{chari-pressley} or \cite{jimbo-86}). Set $\lambda=b/a$. Then
\begin{equation}
	\rRM{a,b}{z}\coloneqq
	\left[
	\begin{array}{cccc}
	1 & 0 & 0 & 0\\
	0 & \frac{q(1-\lambda z)}{q^2-\lambda z} & \frac{\lambda z(q^2-1)}{q^2-\lambda z} & 0\\
	0 & \frac{q^2-1}{q^2-\lambda z} & \frac{q(1-\lambda z)}{q^2-\lambda z} & 0\\
	0 & 0 & 0 & 1
	\end{array}
	\right].
\end{equation}
Note that, if $\lambda=q^{2}$, $\rRM{a,b}{z}$ has a pole at $z=1$, while, if $\lambda=q^{-2}$, $\rRM{a,b}{z}$
is not invertible at $z=1$. It is well--known that $\evrep{1}{a}\ten\evrep{1}{b}$ fails to be
irreducible precisely when $\la=q^{\pm 2}$.
\rmkend
\end{example}

\begin{remark} \label{rmk:Runitarity}
	The R-matrix $\sRMv{VW}{z}$ is in fact a \emph{meromorphic} intertwiner (in the case $q\in\bbC$) \cite{frenkel-reshetikhin-92,kazhdan-soibelman-95}.
	More specifically, relying on the crossing symmetry, \ie the functional relation between $\sRM{VW}{z}$ and $\sRM{V^{**}W}{z}$, one proves that the operator $\sRM{}{z}$ is analytic near zero and therefore extends meromorphically to $\bbC$ (cf.~\cite[Appendix]{etingof-moura-02}).
	\rmkend
\end{remark}

\section{Quantum affine symmetric pairs}\label{s:affine-qsp}

\subsection{Generalized Satake diagrams}\label{ss:gsat}

Classical and quantum Kac-Moody algebras are defined in terms of combinatorial datum encoded by the Dynkin diagram and the Cartan matrix.
Similarly, classical and quantum symmetric pairs \cite{letzter-02,kolb-14} (the latter are also known as {\em quantum symmetric pair (coideal) subalgebras}, {\em Letzter-Kolb coideal subalgebras} and {\em $\iota$quantum groups}, see e.g. \cite{lu-wang-20}) arise from a refinement of such datum.
In the Kac-Moody setting, this relies on the semidirect product decomposition of the automorphism group of the Kac-Moody algebras and the canonical factorization of specific automorphisms given in \cite[4.23 and 4.39]{kac-wang-92} and further developed in \cite[App.~A]{kolb-14} and, somewhat more generally, in \cite{regelskis-vlaar-21}.\\

Let $\Aut(\aff{A})$ be the group of \emph{diagram automorphisms} of the affine
Cartan matrix, \ie the group of bijections $\tau:\aIS\to\aIS$ such that
$a_{ij}=a_{\tau(i)\tau(j)}$.
Let $X \subset\aIS$ be a proper subset of indices.
Note that the corresponding Cartan matrix $A_X$ is necessarily of finite type.
We denote by $\oi_X\in\Aut(A_X)$ the {\em opposition involution of $X$}, \ie the involutive diagram automorphism on $X$ induced by the action of the longest element $w_X$ of the Weyl group $W_X$ on $\Qlat_X$, cf.~\cite[Sec.~3.11]{appel-vlaar-20}. \\

We have the following definition, cf. \cite{regelskis-vlaar-20,regelskis-vlaar-21}.

\begin{definition} \label{def:GSat}
We call a pair $(X,\tau)$ a \emph{generalized affine Satake diagram} and write $(X,\tau)\in\gsat{\aff{A}}$ if $X \subset\aIS$, $\tau$ is an involutive diagram automorphism which preserves $X$, and
\begin{enumerate}
\item $\tau|_X = \oi_X$,
\item for any $i \in \aIS \setminus X$ such that $\tau(i)=i$, the connected component of $X \cup \{ i \}$ containing $i$ is not of type ${\sf A}_2$.
\end{enumerate}
\end{definition}

A classification of generalized Satake diagrams for $\aff{A}$ is given in \cite[App.~A, Tables 5, 6 and 7]{regelskis-vlaar-21}.
Henceforth, we fix $(X,\tau)\in \gsat{\aff{A}}$.

\subsection{Pseudo--involutions}\label{ss:pseudo-involutions}
The diagram automorphism $\tau\in\Aut(\wh{A})$ extends canonically to an automorphism
of ${\agp}$, given on the generators by $\tau(\cort{i})\coloneqq\cort{\tau(i)}$, $\tau(e_i)=e_{\tau(i)}$, and $\tau(f_i)=f_{\tau(i)}$.
We associate to the combinatorial datum $(X,\tau)$ the Lie algebra automorphism $\theta:\agp\to\agp$, given by
\begin{align}\label{def:theta}
	\theta \coloneqq \Ad(w_X) \circ \omega \circ \tau
\end{align}
where $\omega$ denotes the Chevalley involution on $\aff\g$.
Note that, since $\cp{\rt{i}}{\tau(c)}=\cp{\rt{\tau(i)}}{c}=0$,
$\tau$ preserves $\bbC c$. It follows that $\tau(c)=c$ and $\tsat(c)=-c$.
Hence, $\tsat$ descends to an automorphism of $L\fkg$.

\begin{remarks} \label{rmk:KM:pseudoinvolution} \hfill
	\begin{enumerate}[leftmargin=2em]\itemsep0.25cm
	\item
	In \cite[Sec.~4.9]{kac-wang-92}, Kac and Wang defined a canonical procedure (for arbitrary generalized Cartan matrices) to extend a given diagram
	automorphism from ${\ahp}$ to ${\ah}$.
	It follows that $\theta$ canonically extends to an automorphism of ${\ag}$.
	Note that $\tsat$ is an automorphism of ${\ag}$ \emph{of the second kind} (see \cite[4.6]{kac-wang-92}): the two standard nilpotent subalgebras $\fkn^\pm$ are swapped, up to finite-dimensional subalgebras.
	Moreover, it is an involution on ${\ah}$; following \cite{regelskis-vlaar-21}, we will therefore refer to $\tsat$ as a \emph{pseudo--involution of ${\ag}$ of the second kind}.
	We note that ${\ah}^{\theta} \subseteq{\ahp}$, see \cite[Sec.~6.2]{appel-vlaar-20}.

	\item
	Note that the assignment $(X,\tau)\leadsto\tsat$ described in \eqref{def:theta} can be inverted by setting $X = \{ i \in \aIS \, | \, \theta(h_i)=h_i \}$ and subsequently $\tau = \omega \circ \Ad(w_X)^{-1} \circ \theta$.
	Relying on this correspondence, henceforth we shall use the subscript $\tsat$ also for objects explicitly defined in terms of $(X,\tau)$.

	\item
	The dual map $\theta:\ah\to\ah$ is denoted by the same symbol.
	Since $\tsat(c)=-c$, one has $\tsat(\drv{})=-\drv{}$ and hence $\bbZ \drv{}\subseteq \aQ^{-\tsat}$.
	Moreover, $\aQp=\Qp\oplus\bbZ_{\geqslant0}\drv{}$ and therefore $\aQp^{-\tsat}=\Qp^{-\tsat}\oplus\bbZ_{\geqslant0}\drv{}$.
	The \emph{restricted rank} (also known as the relative rank or real rank) of $\tsat$ is the rank of $\aQ^{-\tsat}$ and is equal to the number of $\tau$-orbits in $\wh I \backslash X$ (see e.g.~\cite[Sec.~4]{regelskis-vlaar-21} and \cite[Sec.~8.10]{appel-vlaar-20}).
	It follows that $\tsat$ has restricted rank one if and only if $\aQ^{-\tsat}=\bbZ \drv{}$.
	Suppose $\tsat$ has restricted rank one. For any $j \in \wh I \backslash X$,
	$\theta$ fixes $\al_j-\al_{\tau(j)}$ and each $\al_i$ ($i \in X$), thus yielding
	\eq{ \label{eq:theta:rrank1}
		\theta(\al_j) = \al_j - \frac{2}{a_j \big| \wh I \backslash X \big|} \del
	}
	where we have written $\del = \sum_{i \in \wh I} a_i \al_i$ with $a_0 = 1$.
	\rmkend
\end{enumerate}
\end{remarks}

\subsection{Quantum pseudo--involutions}\label{ss:qtheta}

We shall consider a distinguished lift of the pseudo--involution $\tsat$ to an algebra automorphism $\tsatq$ of $\Uqagp$ and $\UqLg$.
This is obtained by choosing a suitable lift for each of the three factors in $\tsat$.
First, we consider the usual Chevalley involution on $\Uqag$ given by \eqref{eq:chevalley}.
The diagram automorphism $\tau$ extends canonically to an autormorphism of $\Uqagp$ given on the generators by $\tau(\Eg{i})=\Eg{\tau(i)}$, $\tau(\Fg{i})=\Fg{\tau(i)}$ and $\tau(\Kg{h})=\Kg{\tau(h)}$.\\

The action of the Weyl group operator $w_X\in W_X$ is lifted to $\Uqagp$ as follows.
Let $\qWS{X}$ be the braid group operator corresponding to $w_X$ acting on any representation in $\Ointpm(\Uqag)$, cf.~\cite[Sec.~5]{lusztig-94} and also \cite[Sec.~5]{appel-vlaar-20}.
More precisely, given a reduced expression $s_{i_1}\cdots s_{i_{\ell}}$ of $w_X$ in terms of fundamental reflections, $\qWS{X}\coloneqq \qWS{i_1}\cdots\qWS{i_{\ell}}$, where, as in \cite{appel-vlaar-20}, $\qWS{j}=T''_{j,1}$ in the notation from \cite[5.2.1]{lusztig-94}.
It follows from the braid relations that $\qWS{X}$ is independent of the reduced expression.
We consider the Cartan correction of $\qWS{X}$ given by
	\begin{align} \label{def:t-Sat}
		\bt{\tsat} \coloneqq \tcorr{\tsat}\cdot\qWS{X}\, .
	\end{align}
Here $\tcorr{\tsat}$ is the Cartan operator defined on any weight vector of weight $\lambda$ as the multiplication by $q^{\iip{\tsat(\lambda)}{\lambda}/2+\iip{\lambda}{\rho_X}}$, where  $\rho_X$ is the half--sum of the positive roots in $\rootsys_X$ (cf.~\cite[Sec.~4.9]{appel-vlaar-20}).
By \cite[Lemma~4.3 (iii)]{appel-vlaar-20}, we obtain an automorphism $\adbt{\tsat}: \Uqagp\to\Uqagp$ given by $\adbt{\tsat}\coloneqq\Ad{\bt{\tsat}}$.\\

The \emph{quantum pseudo--involution} is the automorphism $\tsatq:\Uqagp\to\Uqagp$ given by
\begin{equation} \label{def:thetaq}
\tsatq\coloneqq\adbt{\tsat}\circ\omega\circ\tau.
\end{equation}
Note that, as in the classical case, $\tsatq$ is independent of the order of the three factors.
In \cite[Lem.~6.10]{appel-vlaar-20}, we derive the following important properties of $\tsatq$:
\begin{align}
\label{theta:Cartan} \tsatq(K_h) &= K_{\theta(h)} && \text{for all } h \in \aQvext, \\
\label{theta:rootspace} \tsatq\big((U_q\wh\fkg)_\la \big) &= (U_q\wh\fkg)_{\theta^*(\la)} && \text{for all } \la \in \aQ, \\
\label{theta:X} \tsatq|_{\Uqag_X} &= \id_{\Uqag_X},
\end{align}
where for all $X \subset \wh I$ we have introduced the subalgebra $\Uqagp_X := \langle E_i, F_i, K_i^{\pm 1} \rangle_{i \in X}$ of $\Uqagp$, which is a Drinfeld-Jimbo quantum group of finite type.

\begin{remark}
Further to Remark \ref{rmk:KM:pseudoinvolution}, note that the Kac-Wang extension of the diagram automorphism $\tau$ to ${\ah}$ does not necessarily preserve the extended coroot lattice $\aQvext\subset{\ah}$, and therefore does not automatically extend to an automorphism of $\Uqag$.
To remedy this, one can modify the lattice itself by replacing the standard derivation $\codrv{}\in{\ah}$ with any $\codrv{\tau}\in{\ah}$ such that  $\rt{i}(\codrv{\tau})=\rt{\tau(i)}(\codrv{\tau})$, see \cite[Sec.~2.6]{kolb-14}. \rmkend
\end{remark}

For the purposes of this paper it suffices to regard $\tsatq$ as an automorphism of $\Uqagp$.
Furthermore, $\tsatq$ descends to an automorphism of the quantum loop algebra $\UqLg$.

\subsection{QSP subalgebras}\label{ss:qsp}
By \cite[Sec.~6.2]{appel-vlaar-20}, ${\ah}^{\tsat}\subset{\ahp}$, thus
one obtains $(U_q{\ah})^{\tsatq} = U_q({\ah}^\tsat) \subseteq U_q{\ahp}$.
To a given pseudo--involution $\tsat$ of the second kind, we associate a family of coideal subalgebras of $\Uqagp$ intersecting $\Uqahp$ in $(U_q{\ah})^{\tsatq}$.
This family is parametrized by two sets,
$\Parsetc\subset(\bsF^\times)^{\aIS}$ and $\Parsets\subset\bsF^{\aIS}$, introduced in \cite{letzter-02,kolb-14,regelskis-vlaar-20,regelskis-vlaar-21}.

\begin{definition}
The {\em QSP subalgebra} of $\Uqagp$ corresponding to $\tsat$ with parameters  $(\Parc,\Pars)\in \Parsetc \times \Parsets$ is the subalgebra
	\begin{align}
		\Uqk \coloneqq \Uqk_{\,\tsat,\parc{},\pars{}}
		=\langle \Uqag_X, (U_q{\ahp})^{\tsatq}, \Bg{i}\, | \, i\in \aIS \setminus X \rangle \, ,
	\end{align}
where the elements $\Bg{i} := \Bg{i;\, \tsat,\Parc_i,\Pars_i}\in\Uqagp$ are given by
\eq{ \label{Bi:def}
	\Bg{i} :=
	\left\{
	\begin{array}{cl}
		\Fg{i} & \mbox{if }\, i\in X\, ,\\
		\Fg{i} + \parc{i}\cdot\tsatq(\Fg{i}) + \pars{i}\cdot\Kg{i}^{-1}  & \mbox{if }\, i\not\in X\, .
	\end{array}
	\right.
}
\end{definition}
\begin{remark} \label{rmk:parameters}
More precisely, $\Parsetc$ and $\Parsets$ are defined as follows, see \cite[Eqns.~(5.9) and (5.11)]{kolb-14}.
First, we choose a subset $\aIS^* \subseteq \aIS \setminus X$ by picking a representative for every $\tau$-orbit in $\aIS\setminus X$.
Then, we define
\begin{align}
	\aIdiff &= \{ i \in \aIS^* \, | \, \tau(i) \ne i \text{ and } \exists j \in X \cup \{ \tau(i)\} \text{ such that } a_{ij} \ne 0 \}, \\
	\aIns &= \{ i \in \aIS^* \, | \, \tau(i) =i \text{ and } \forall j \in X \, a_{ij} = 0 \}.
\end{align}
Now $\Parsetc$ is the set of tuples $\Parc\in(\bsF^\times)^{\aIS}$ such that $\parc{i}=1$ if $i\in X$ and $\parc{i} = \parc{\tau(i)}$ if $\{i , \tau(i) \} \cap \aIdiff = \emptyset$.
Furthermore, $\Parsets$ is the set of tuples $\Pars \in \bsF^{\aIS}$ such that $\pars{i}=0$ if $i\in\aIS\setminus\aIns$ and, for all $(i,j) \in \aIns^{\times 2}$, $a_{ij} \in 2\bbZ$ or $\Pars_j = 0$.
Note that $\Parsetc$ and $\Parsets$ do not depend upon the choice of $\aIS^*$.
The constraints on the parameter sets $\Parsetc$ and $\Parsets$ are motivated by Proposition~\ref{prop:parameter-coideal} below.
\hfill \rmkend
\end{remark}

Note that $\Uqagp_X (U_q{\ahp})^{\tsatq} \subset\Uqk$ is a subbialgebra of $\Uqagp$.
The following result follows directly from \cite[Props.~5.2 and 6.2]{kolb-14}\footnote{Kolb's proofs straightforwardly extend to the case of generalized Satake diagrams.}.

\begin{prop}\label{prop:parameter-coideal}
For any pseudo-involution of the second kind $\theta$, $\Uqk$ is a right coideal subalgebra of $\Uqagp$. Moreover, whenever $(\parc{},\pars{})\in\Parsetc\times\Parsets$, $\Uqk$
has minimal intersection with $\Uqahp$:
\eq{
\Delta(\Uqk)\subset\Uqk\ten\Uqagp, \qq \qq \Uqk\cap\Uqahp = (\Uqahp)^{\tsatq}.
}
\end{prop}

\begin{remark}\label{rmk:parameter-operator}
Following \cite[Sec.~7.4]{appel-vlaar-20}, we shall regard the tuple $\Parc$ as a diagonal operator on category $\cO^+$ representations.
Namely, we fix henceforth a group homomorphism $\parc{}:\aP\to\bbF^\times$ such that $\parc{}(\rt{i})\coloneqq\parc{i}$ ($i\in\aIS$).
Then, $\parc{}$ acts on any weight vector of weight $\lambda$ as multiplication by $\parc{}(\lambda)$.
\end{remark}

\subsection{The standard K-matrix}\label{ss:univ-kmx}

In \cite[Thm.~8.11-8.12]{appel-vlaar-20}, the authors proved that the QSP
subalgebra $\Uqk$ gives rise to a discrete family of K-matrices in $\Uqag$,
indexed by affine generalized Satake diagrams. In Section~\ref{ss:gauging-k},
we shall provide an even more general construction. We first recall the definition of the
\emph{standard K-matrix}.

\begin{theorem}[\cite{appel-vlaar-20}]\label{thm:av-k-mx}
	There exists a unique series $\QK{\tsat}\coloneqq 1+\sum_{\mu \in\aQp^{-\tsat}}\QK{\tsat,\mu}$
	with $\QK{\tsat,\mu}\in\Uqanp_\mu$ such that the operator
	$\KM{\tsat}\coloneqq\Parc^{-1}\cdot\QK{\tsat}\in\CUqag{}$
	satisfies the intertwining identity
	\begin{align}\label{eq:std-k-intertwiner}
		\KM{\tsat}\cdot b=\tsatq^{-1}(b)\cdot\KM{\tsat}  \qquad(b \in \Uqk)
	\end{align}
	and the coproduct identity
	\begin{align}\label{eq:std-coprod-id}
		\Delta(\KM{\tsat})=\RM{\tsat}^{-1}\cdot 1\ten\KM{\tsat}
		\cdot\RM{}^{\tsatq^{-1}}\cdot\KM{\tsat}\ten 1
	\end{align}
	where $\RM{}$ is the $R$--matrix of $\Uqag$, $\RM{}^{\tsatq^{-1}} \coloneqq\tsatq^{-1}\ten \id(R) $, and $\RM{\tsat}\coloneqq(\bt{\tsat}\ten\bt{\tsat})^{-1}\cdot \Delta(\bt{\tsat})$ is (up to a Cartan correction)
	the R-matrix of $\Uqg_X$.
	Moreover, a constant version of the generalized reflection equation holds:
	\begin{align}\label{eq:std-tw-RE}
		\RM{21}^{\tsatq^{-1}\tsatq^{-1}}\cdot 1\ten\KM{\tsat}
		\cdot\RM{}^{\tsatq^{-1}}\cdot\KM{\tsat}\ten 1
		=
		\KM{\tsat}\ten 1\cdot(\RM{}^{\tsatq^{-1}})_{21}\cdot 1\ten\KM{\tsat}
		\cdot\RM{}.
	\end{align}
\end{theorem}

\begin{remark}
We shall refer to the operator $\QK{\tsat}$ as \emph{the quasi K-matrix} of $\Uqk$, although in terms of the quasi K-matrix $\mathfrak{X}_{\tsat}$ defined in \cite{bao-wang-18b} and \cite{balagovic-kolb-19} as the unique intertwiner between the two bar involutions on $\Uqag$ and
$\Uqk$ we have $\QK{\tsat} = \ol{\mathfrak{X}_{\tsat}}$.
Moreover, our construction in \cite{appel-vlaar-20} of $\QK{\tsat}$ does not rely at all on the QSP bar involution.
In fact, as later observed in \cite{kolb-21}, $\QK{\tsat}$ can be used to \emph{define} the bar involution on $\Uqk$.	\rmkend
\end{remark}

\subsection{Gauging K-matrices}\label{ss:gauging-k}\label{ss:distinguished-k}

New solutions of the generalized reflection equations are easily obtained by
acting simultaneously on the K-matrix $\KM{\tsat}$ and the operator $\tsatq^{-1}$.
Consider the following group:
\eq{
\Gg\coloneqq \Big\{\gau\in \big( \CUqag{} \big)^\times \;\Big\vert\; \Ad(\gau)(\Uqagp)=\Uqagp,\, \varepsilon(\gau)=1 \Big\}.
}
The condition $\varepsilon(\gau)=1$ is a non-essential convenient normalization condition.
We have the following
\begin{corollary}\label{cor:av-k-mx}
	Let $\gau\in\Gg$ and set $\psi\coloneqq\Ad(\gau)\circ\tsatq^{-1}$ and
	$\RM{\psi}\coloneqq(\gau\bt{\tsat}\ten \gau\bt{\tsat})^{-1}\Delta(\gau\bt{\tsat})$
	The operator
	\begin{equation}\label{eq:universal-K-matrix}
	\KM{\psi}\coloneqq \gau \cdot\Parc^{-1}\cdot\QK{\tsat}\in\CUqag{}
	\end{equation}
	satisfies the intertwining identity
	\begin{align}\label{eq:k-intertwiner}
		\KM{\psi}\cdot b=\psi(b)\cdot\KM{\psi} \qquad(b \in \Uqk)
	\end{align}
	and the coproduct identity
	\begin{align}\label{eq:coprod-id}
		\Delta(\KM{\psi})=\RM{\psi}^{-1}\cdot 1\ten\KM{\psi}
		\cdot\RM{}^{\psi}\cdot\KM{\psi}\ten 1
	\end{align}
	where $\RM{}$ is the $R$--matrix of $\Uqag$ and $\RM{}^{\psi} \coloneqq\psi\ten \id(R) $. Moreover, the generalized reflection equation holds:
	\begin{align}\label{eq:tw-RE}
		\RM{21}^{\psi\psi}\cdot 1\ten\KM{\psi}
		\cdot\RM{}^{\psi}\cdot\KM{\psi}\ten 1
		=
		\KM{\psi}\ten 1\cdot(\RM{}^{\psi})_{21}\cdot 1\ten\KM{\psi}
		\cdot\RM{}.
	\end{align}
\end{corollary}
\begin{proof}
It is enough to observe that, under the identity $\KM{\psi}=\gau\cdot\KM{\tsat}$, the equations
\eqref{eq:k-intertwiner}, \eqref{eq:coprod-id}, and \eqref{eq:tw-RE} reduce to their analogues
for the standard K-matrix $\KM{\tsat}$ in Theorem~\ref{thm:av-k-mx}.
\end{proof}
As before, the evaluation of $\KM{\tsat}$ on $V\in\Ointp(\Uqag)$
yields a QSP intertwiner
\begin{equation}
	\KM{\psi, V}:V\to\psi^*(V)
\end{equation}

We shall refer to the automorphism $\psi$ as the \emph{twisting operator} of
the reflection equation.
The coproduct identity \eqref{eq:coprod-id} and the reflection
equation \eqref{eq:tw-RE} both admit a similar representation theoretic
interpretation. Namely, note that, for any $U\in\Rep{\Uqagp}$ and $W\in\Ointp(\Uqagp)$,
the operator $\RM{UW}\in\End(U\ten W)$  is well-defined\footnote{
Note however that this is not true for $\RM{WU}$.}, and therefore so is
$\RM{\psi^*(V)W}$.
Moreover, by construction, one has
\begin{align}
	\RM{21}^{\psi\psi}=\RM{\psi, 21}\cdot \RM{}\cdot \RM{\psi}^{-1}\,,
\end{align}
The operator $\RM{\psi^*(V)\psi^*(W)}$ is also well-defined and \eqref{eq:tw-RE} holds.
As mentioned in the statement of the Theorem, the latter is a constant (\ie parameter-independent) version of the \emph{generalized reflection equation}  introduced by Cherednik in \cite[Eq.~(4.14)]{cherednik-92} (see also \cite{freidel-maillet-91}), which encodes the quartic relations in the cylindrical braid group, cf.~\cite[Sec.~2]{appel-vlaar-20}\footnote{
More precisely, this yields an action of a cylindrical ribbon braid {\em groupoid}, since the involved
operators cannot always be arbitrarily composed. However, in special cases, this action does yields
a representation of the cylindrical braid group, \eg if $\psi^*(V)=V$ and $\psi^*(W)=W$.
}.
Note that \eqref{eq:tw-RE} depends on the choice of $\psi$.

\begin{remark}\label{rmk:distinguished}
We describe several distinguished examples of $\gau \in \Gg$ and the corresponding K-matrices and twists.

	\begin{enumerate}\itemsep0.25cm
	\item First of all, we may consider diagonal modifications of the K-matrix obtained by setting $\gau={\bm \beta}$, where ${\bm \beta}$ is
	any map $\aP\to\bsF^{\times}$.

	\item If $\gau=\bt{\tsat}$ then the twisting operator is the involution $\psi=\chev\circ\tau$.
	We refer to the corresponding K-matrix as the \emph{semi-standard} K-matrix.

	\item Let $(Y,\eta)\in\gsat{\aff{A}}$ be a Satake diagram and $\zsat$ the corresponding pseudo-involution.
	For $\gau=\bt{\zsat}^{-1}\bt{\tsat}$, one recovers the combinatorial family of universal K-matrices constructed in \cite{appel-vlaar-20}.
	There are two important special cases:
		\begin{itemize}
		\item If $\zsat=\tsat$, we get the standard K-matrix.
		\item The second special case is obtained in analogy with {\em the} finite-type K-matrix constructed by
		Balagovi\'{c} and Kolb in \cite{balagovic-kolb-19}, which corresponds to the choice\footnote{
				Note that $(\fIS,\oi_{\fIS})$ can be regarded as a degenerate case, since the corresponding QSP subalgebra is the full
				quantum group $\Uqg$. This choice is only available in finite type.}
		$(Y, \eta) = (\fIS,\oi_{\fIS})$, whose associated pseudo-involution is $\ze = \id$.
		The affine analogue of this choice, described in \cite[Sec.~9]{appel-vlaar-20}, is obtained
		by taking $Y=\aIS\setminus\{0,\tau(0)\}$, $\eta(0)=\tau(0)$, and $\eta|_{Y}=\oi_{Y}$.
		Casework shows that $\eta$ thus defined is a diagram automorphism, so $(Y,\eta)$ is an affine Satake diagram.	 \rmkend
		\end{itemize}

	\end{enumerate}
\end{remark}

\section{Spectral K-matrices}\label{s:spectral}

In this section, we present the first main result of the paper.
Let $(X,\tau)\in\gsat{\aff{A}}$ with pseudo-involution $\tsat$, $(\parc{},\pars{})\in\Parsetc\times\Parsets$, and $\Uqk\subset\Uqagp$ the  corresponding QSP subalgebra.
We prove that the universal K-matrices constructed in \cite{appel-vlaar-20} and Corollary~\ref{cor:av-k-mx} specialize under mild assumptions to spectral operators on finite-dimensional $\UqLg$-representations.

\subsection{$\tau$-minimal grading shifts}\label{ss:tau-grading}
We shall need to replace the homogeneous grading shift defined in \ref{ss:spectral-R}, most
commonly used in the context of quantum loop algebras, with a distinguished
$\tau$-invariant grading shift. Note that for any group homomorphism $s: \aQ \to \Z$, there is
a corresponding grading shift $\Sigma^s_z:\UqLg\to\UqLg[z,z^{-1}]$ defined by
\eq{
	\Sigma^s_z(E_i) = z^{s(\al_i)} E_i, \qq \Sigma^s_z(F_i) = z^{-s(\al_i)} F_i, \qq \Sigma^s_z(K_h) = K_h
}
for $i \in \wh I$, $h \in \aQv$. Then, $\Sigma^s_z$ is \emph{$\tau$-invariant} if $s \circ \tau = s$, \ie $\Sigma^s_z \circ \tau = \tau \circ \Sigma^s_z$. Note that $s$ is $\tau$-invariant if and only if, as a function on the set of affine simple roots $\{ \al_i \}_{i \in \wh I}$, $s$ is the characteristic function of a union of $\tau$-orbits. In particular, the principal grading shift $\shift{z}^{\sf pr}$ defined by $s(\rt{i})=1$ ($i\in\aIS$) is always $\tau$-invariant.
The \emph{$\tau$-minimal} grading shift $\Sigma^{s_\tau}_z$ corresponds instead
to the characteristic function $s_\tau$ of the $\tau$-orbit of the affine node $0$, \ie
\eq{
	s_\tau(\al_i) = \begin{cases}
		1 & \text{if } i \in \{ 0,\tau(0) \}, \\
		0 & \text{otherwise}
	\end{cases}
}
Then, $\shift{z}^{s_\tau} = \shift{z}^{\sf hom}$ if and only if $\tau(0)=0$.
Note that the analogue of Theorem~\ref{thm:spectral-R} holds if we replace $R(z)$ by $\id \ten \Sigma^{s_\tau}_z(R)$ and $\Delta_z$ by $\id\ten\Sigma^{s_\tau}_z \circ \Delta$.\\

From now one we shall use the $\tau$-minimal grading shift $\Sigma_z^{s_\tau}$ and drop the
upper index $s_{\tau}$ unless needed. In particular,  we shall denote by $\pi_{V,z}$ the action
on $V$ shifted by $\Sigma_z^{s_{\tau}}$.

\subsection{Spectral K-matrices}\label{ss:spectral-k}

Let $\GQSP\subset\cG$ be the subset of gauge transformations $\gau\in\cG$
of the form $\gau\coloneqq S_Y^{-1}S_X\cdot {\bm \beta}$, where
\begin{enumerate}\itemsep0.25cm
	\item $Y\subset\aff{I}$ is any proper subdiagram such that $\shifta{i}=0$ for any $i\in Y$;
	\item ${\bm \beta}:\aP\to\bsF^\times$ is any map such that ${\bm \beta}(\drv{})=\parc{}(\drv{})$.
\end{enumerate}
About the definition of the set $\GQSP$ see also Remark~\ref{rmk:rational}(2).\\

We refer to a twisting operator of the form $\psi=\Ad(\gau)\circ\tsat_q^{-1}$ with $g\in\GQSP$, as a {\em QSP-admissible}
twisting operator.
We shall prove the following spectral analogue of Theorem~\ref{thm:av-k-mx} and Corollary~\ref{cor:av-k-mx}.

\begin{theorem}\label{thm:spectral-k}
	The quantum loop algebra $\UqLg$ has a $\GQSP$-family of universal
	\emph{spectral} K-matrices relative to the QSP subalgebra $\Uqk$. More precisely,
	for any $\gau\in\GQSP$, set
	\begin{equation}
		\psi\coloneqq\Ad(\gau)\circ\tsat_{q}^{-1} \quad\mbox{and}\quad
		\RM{\psi}\coloneqq(\bt{\psi}\ten\bt{\psi})^{-1}\cdot\Delta(\bt{\psi})
	\end{equation}
	where $\bt{\psi}\coloneqq \gau\cdot\bt{\tsat}$.
	There exists a canonical Laurent series $\sKM{\psi}{z}\in\Lfml{\CUqLg{}}{z}$
	such that $\shift{a}(\sKM{\psi}{z})=\sKM{\psi}{az}$ ($a\in\bsF^\times$) and the following properties hold.
	\vspace{0.25cm}
	\begin{enumerate}\itemsep0.25cm
		\item For any $b\in\Uqk$,
		\begin{align}\label{eq:spectral-k-intertwiner}
			\sKM{\psi}{z}\cdot\shift{z}(b)=\psi(\shift{1/z}(b))\cdot\sKM{\psi}{z}\, .
		\end{align}
		\item Set $\sRM{}{z}^{\psi} \coloneqq\psi\ten \id(\sRM{}{z}) $. Then,
		\begin{align}\label{eq:spectral-k-coproduct}
			\Delta_{w/z}(\sKM{\psi}{z})=\RM{\psi}^{-1}\cdot 1\ten\sKM{\psi}{w}
			\cdot\sRM{}{zw}^{\psi}\cdot\sKM{\psi}{z}\ten 1\, .
		\end{align}
	\end{enumerate}
	\vspace{0.25cm}
	Moreover, $\sKM{\psi}{z}$ is a solution of the generalized reflection equation
	\begin{align}\label{eq:spectral-tw-RE}
		\sRM{}{w/z}_{21}^{\psi\psi}\cdot 1\ten\sKM{\psi}{w}
		\cdot&\sRM{}{zw}^{\psi}\cdot\sKM{\psi}{z}\ten 1=\\
		&=\sKM{\psi}{z}\ten 1\cdot\sRM{}{zw}_{21}^{\psi}\cdot 1\ten\sKM{\psi}{w}
		\cdot\sRM{}{w/z}\, ,
	\end{align}
	where $\sRM{}{z}_{21}^{\psi}\coloneqq\psi\ten\id(\sRM{}{z})_{21}$.
\end{theorem}

\begin{remark}\label{rmk:spectral-k}\hfill
\begin{enumerate}\itemsep0.25cm
\item
The identities \eqref{eq:spectral-k-intertwiner} and \eqref{eq:spectral-k-coproduct} hold in $\Lfml{\CUqLg{\ten 2}}{w/z,z}$,
where $\Lfml{\bsF}{w/z,z}\coloneqq\operatorname{Frac}(\fml{\bsF}{w/z,z})$.
Following \cite[Eq.~(10)]{cherednik-84}, it may be convenient to use an \emph{adapted} set of ``simple root'' coordinates, given
by $u=w/z$ and $v=z$. Then, \eqref{eq:spectral-k-intertwiner} and \eqref{eq:spectral-k-coproduct} read
\begin{align}
	\Delta_{u}(\sKM{\psi}{v})=\RM{\psi}^{-1}\cdot 1\ten\sKM{\psi}{uv}
	\cdot\sRM{}{uv^2}^{\psi}\cdot\sKM{\psi}{v}\ten 1\, ,
\end{align}
and
\begin{align}
	\sRM{}{u}_{21}^{\psi\psi}\cdot 1\ten\sKM{\psi}{v}
	\cdot&\sRM{}{uv^2}^{\psi}\cdot\sKM{\psi}{uv}\ten 1=\\
	=&\sKM{\psi}{uv}\ten 1\cdot\sRM{}{uv^2}_{21}^{\psi}\cdot 1\ten\sKM{\psi}{v}
	\cdot\sRM{}{u}\, ,
\end{align}
in $\Lfml{\CUqLg{\ten 2}}{u,v}$.
\item
Whenever $\gau$ is shift-invariant, \ie $\shift{z}(\gau)=\gau$, the K-matrix
$\sKM{\psi}{z}$ is a formal series in $\fml{\CUqLg{}}{z}$.
\item
The QSP subalgebra of $U_qL\fksl_2$ for which $X=\emptyset$ and $\tau$ is the nontrivial diagram automorphism is also known as the \emph{augmented q-Onsager algebra}.
In \cite[Sec. 4.1.2]{baseilhac-tsuboi-18} so-called \emph{generic} K-operators in a completion of $\langle K_1 \rangle \otimes \bbF(z)$ are considered.
It would be interesting to relate these to the spectral universal K-matrix $K_{\theta_q^{-1}}(z)$.
\rmkend
\end{enumerate}
\end{remark}

In analogy with the case of the R-matrix, the spectral K-matrix is obtained by applying the shift
operator to the universal K-matrix $\KM{\psi}$ from Theorem~\ref{thm:av-k-mx},
\ie $\sKM{\psi}{z}\coloneqq\shift{z}(\KM{\psi})$.
The identities \eqref{eq:spectral-k-intertwiner}, \eqref{eq:spectral-k-coproduct}, and \eqref{eq:spectral-tw-RE}
are then recovered from their analogues \eqref{eq:k-intertwiner}, \eqref{eq:coprod-id}, and \eqref{eq:tw-RE}, respectively,
by applying the shift operator $\shift{z}\ten\shift{w}$. Clearly, since the operator $\sKM{\psi}{z}$ is valued in
$\CUqLg{}$, the statements above are to be interpreted as operators on finite-dimensional representations in $\Repfd(\UqLg)$ and it is therefore necessary to prove that $\sKM{\psi}{z}$ gives rise to a well defined element
$\sKM{\psi,V}{z}\in\Lfml{\End(V)}{z}$ for any $V\in\Repfd(\UqLg)$.
The proof is carried out in Sections \ref{ss:k-fd-rep}-\ref{ss:k-fd-rep-2}.

\subsection{Descent to finite-dimensional representations}\label{ss:k-fd-rep}
The first step in the proof of Theorem~\ref{thm:spectral-k} amounts to proving that,
for any $V\in\Repfd(\UqLg)$ with representation $\pi_V:\UqLg\to\End(V)$, we obtain a
well-defined operator
\begin{equation}
	\sKM{\psi,V}{z}\coloneqq\pi_{V,z}(\KM{\psi})=\pi_{V}\circ\shift{z}(\KM{\psi})
	\in\Lfml{\End(V)}{z}
\end{equation}
More precisely, we shall prove that each of the operators involved in the definition of the universal K-matrix \eqref{eq:universal-K-matrix} and in the coproduct identity \eqref{eq:spectral-k-coproduct} descends to one on any finite-dimensional representation over $\UqLg$.  We shall need the following

\begin{prop}\label{prop:descent}
The following operators in $\CUqag{}$ descend to $\CUqLg{}$:
\begin{enumerate}\itemsep0.25cm
	\item the operator $\tcorr{\tsat}$, defined on any weight vector $v$ of weight $\lambda$ by
	 \begin{equation}
	 	\tcorr{\tsat}(v)=q^{\iip{\tsat(\lambda)}{\lambda}/2+\iip{\lambda}{\rho_X}}\cdot v\, ,
	 \end{equation}
	where $\rho_X$ is the half-sum of the positive roots in $\rootsys_X$ (cf.~\ref{ss:qtheta});
	\item for any $i\in\aIS$, the braid group operator $\qWS{i}$;
	\item for any map $\bm{\psi}:\aP\to\bbF^\times$ such that $\bm{\psi}(\drv{})=1$, the diagonal operator defined on any weight vector $v$ of weight $\lambda$ by $\bm{\psi}(v)\coloneqq\bm{\psi}(\lambda)\cdot v$.
\end{enumerate}
Moreover,
\begin{enumerate}\itemsep0.25cm
	\item[(4)] for any $\Psi\in\wh{\bigoplus}_{\mu\in\aQp}\Uqanp_{\mu}$, the shifted operator $\shift{z}(\Psi)$ descends
	to $\fml{\CUqLg{}}{z}$.
\end{enumerate}
\end{prop}

\begin{proof}
(1) In analogy with the operator $q^{\wh{\Omega}_0}$ from \ref{ss:R-matrix}, one checks
easily that
\begin{align}
\tcorr{\tsat}=q^{\sum_{i\in\fIS} \tsat(u^i)u_i+m(\tsat(c)d+\tsat(d)c)-\rho^\vee_{X}}\, ,
\end{align}
where $m=1,2,3$ if $\g$ is of type $\mathsf{ADE}$, $\mathsf{BCF}$, or $\mathsf{G}$, respectively.
Since $\tsat(c)=-c$, the term $m(\tsat(c)d+\tsat(d)c) = m(\tsat(d)-d)c$ acts as 0 and can be ignored.
Therefore, as in the case of the operator $q^{\wh{\Omega}_0}$ in \ref{ss:spectral-R}, $\tcorr{\tsat}$
descends to an operator in $\CUqLg{}$. Note that $\tcorr{\tsat}(\drv{})=1$.\\

(2) By restriction, $V$ is a finite-dimensional representation of $U_q\wh\fkg_{\{i\}}$ and therefore integrable.
In particular, the action of $\qWS{i}$ on $V$ is well-defined.\\

(3) By definition, a type $\bf{1}$ representation $V$ admits a weight decomposition over
the quotient lattice $\aPd$, \ie $V=\bigoplus_{\lambda\in\aPd} \wsp{V}{\lambda}$.
Therefore, the operator $\bm{\psi}$ acts on $V$ if and only if
it factors through $\aPd$, \ie if $\bm{\psi}(\drv{})=1$.\\

(4) Set $\Psi=\sum_{\mu\in\aQp}\Psi_{\mu}$, with $\Psi_{\mu}\in\Uqanp_{\mu}$.
Let $p:\aQp\to\bbZ_{\geqslant0}$ be the evaluation map given by $p(\lambda)\coloneqq\lambda(\codrv{\shift{}})$,
so that $\shift{z}(\Psi)=\sum_{n\geqslant0}\Psi(n)z^n$ where $\Psi(n)\coloneqq\sum_{\mu\in p^{-1}(n)}\Psi_{\mu}$. Fix $n\geqslant0$. We shall prove that $\Psi(n)$ is a well-defined
operator on $V$. Note that, in the case of the principal grading shift (cf.~\ref{ss:tau-grading}),
$p^{-1}(n)$ is a finite set and the result is clear.
Assume without loss of generality that $\shifta{0}\neq0$.
Any $\mu\in p^{-1}(n)$ has the form $\mu=m\rt{0}+\lambda$, where $m\leqslant n$ and $\lambda\in\Qp$. However, the action of
$\Uqnp$ is locally finite on $V$ and the number of occurrences of $\Eg{0}$ is bounded by $n$.
Therefore, the action of $\Psi_{\mu}$ on $V$ is non-zero only for finitely many $\mu\in p^{-1}(n)$
and $\Psi(n)$ is well-defined on $V$. The same argument applies if $0$ is replaced by any other
node $i\in\aIS$ such that $\shifta{i}\neq0$.
\end{proof}

By Theorem~\ref{thm:av-k-mx} and \eqref{eq:universal-K-matrix}, we have
	\begin{align}
	\KM{\psi}=\gau \cdot\Parc^{-1}\cdot\QK{\tsat}\in\CUqag{}
\end{align}
where we recall that $\gau\in\GQSP$ is a gauge, $\Parc{}$ is  the parameter operator, and
$\QK{\tsat}$ is the quasi-K-matrix.
Note that, by definition of $\GQSP$, $\shift{z}(\gau\cdot\Parc{})$ is an element of $\CUqLg{}[z,z^{-1}]$.
Therefore, by the result above, $\sKM{\psi}{z}\coloneqq\shift{z}(\KM{\psi})\in\Lfml{\CUqLg{}}{z}$, \ie for any $V\in\Repfd(\UqLg)$, $\sKM{\psi,V}{z}\coloneqq \pi_{V,z}(\KM{\psi})$ is a well-defined operator in $\Lfml{\End(V)}{z}$.

\begin{remark}\label{rmk:special-formal-cases}
	We did not use yet the fact that the grading shift is $\tau$--invariant and
	$\gau\cdot\bt{\tsat}$ is shift-invariant. These assumptions become
	crucial in the next step.
	\rmkend
\end{remark}

\subsection{Spectral K-matrices on finite-dimensional representations}\label{ss:k-fd-rep-2}
We complete the proof of Theorem~\ref{thm:spectral-k} by showing that the relations
\eqref{eq:spectral-k-intertwiner}, \eqref{eq:spectral-k-coproduct}, and \eqref{eq:spectral-tw-RE} hold.
Note that, since the grading shift is $\tau$-invariant and $\gau\in\GQSP$ is shift-invariant,
we have $\tau\circ\shift{z}=\shift{z}\circ\tau$ and $\Ad(\gau\cdot\bt{\tsat})\circ\shift{z}=\shift{z}\circ\Ad(\gau\cdot\bt{\tsat})$.
Finally, since $\chev\circ\shift{z}=\shift{1/z}\circ\chev$ and $\psi=\Ad(\gau\cdot\bt{\tsat})\circ\chev\circ\tau$, we have
\begin{align}\label{eq:parameter-inversion}
\psi\circ\shift{z}=\shift{1/z}\circ\psi\, .
\end{align}
Let $V\in\Repfd(\UqLg)$.
The action on the shifted representation $\Lfml{\psi^*(V)}{1/z}$ is, by definition, given by
\begin{equation}\label{eq:pullback}
	\pi_{\psi^*(V), 1/z}(x)=\pi_{\psi^*(V)}(\shift{1/z}(x))=\pi_V(\psi\circ\shift{1/z}(x))\, .
\end{equation}
Since $\pi_V(\psi\circ\shift{1/z}(x))=\pi_V(\shift{z}\circ\psi(x))$, one has
$\Lfml{\psi^*(V)}{1/z}=\psi^*(\Lfml{V}{z})$.
Therefore, the evaluation of the intertwining identity \eqref{eq:k-intertwiner}
on $\Lfml{V}{z}$ through $\pi_{V,z}$ yields
\begin{align}\label{eq:spectral-k-intertwining-rep}
\sKM{\psi,V}{z}\pi_{V,z}(b)=\pi_{\psi^*(V), 1/z}(b)\sKM{\psi,V}{z}\, .
\end{align}
It follows that $\sKM{\psi,V}{z}$ is a $\Uqk$-intertwiner $\Lfml{V}{z}\to\Lfml{\psi^*(V)}{1/z}$,
which is equivalent to \eqref{eq:spectral-k-intertwiner}.\\

Set $\sKM{\psi,VW}{z,w}\coloneqq\pi_{V,z}\ten\pi_{W,w}(\Delta(\KM{\psi}))$.
Then, the evaluation of the
coproduct identity \eqref{eq:coprod-id} on $\Lfml{V}{z}\ten\Lfml{W}{w}$ through $\pi_{V,z}\ten\pi_{W,w}$ yields
\begin{align}\label{eq:spectral-k-coproduct-rep}
		\sKM{\psi,VW}{z,w}=\RM{\psi, VW}^{-1}\cdot \id\ten\sKM{\psi, W}{w}
\cdot\sRM{\psi^*(V)W}{zw}\cdot\sKM{\psi, V}{z}\ten \id\, ,
\end{align}
By \eqref{eq:pullback}, this is equivalent to \eqref{eq:spectral-k-coproduct}.
Finally, the evaluation of  the generalized twisted reflection equation
\eqref{eq:tw-RE} on $\Lfml{V}{z}\ten\Lfml{W}{w}$ through $\pi_{V,z}\ten\pi_{W,w}$
yields
\begin{align}\label{eq:spectral-tw-RE-rep}
	\sRM{\psi^*(W)\, \psi^*(V)}{w/z}_{21}&\cdot \id\ten\sKM{\psi, W}{w}
	\cdot\sRM{\psi^*(V)W}{zw}\cdot\sKM{\psi, V}{z}\ten \id=\\
	&=\sKM{\psi, V}{z}\ten \id\cdot\sRM{\psi^*(W)\, V}{zw}_{21}\cdot \id\ten\sKM{\psi,W}{w}
	\cdot\sRM{VW}{w/z}\, ,
\end{align}
where $\sRM{WV}{z}_{21}\coloneqq(1\, 2)\circ\sRM{WV}{z}\circ (1\, 2)$.
As before, this is equivalent to \eqref{eq:spectral-tw-RE}.

\section{Rational K-matrices}\label{s:rational}

In this section, we prove that, in the case of irreducible representations,
the spectral K-matrix constructed in Theorem~\ref{thm:spectral-k}
gives rise to rational solutions of the generalized reflection equation.

\subsection{Rational QSP intertwiners}\label{ss:rational-intertwiners}
Theorem~\ref{thm:spectral-k} leads to the existence of a rational QSP intertwiner\footnote{Since the QSP subalgebra $\Uqk$ is fixed throughout this Section, we shall use the expression {\em QSP intertwiner} instead of {\em $\Uqk$-intertwiner}.}, as follows\footnote{The authors are grateful to V. Toledano Laredo for pointing out this argument.}.

\begin{lemma}\label{lem:rational-intertwiner}
Let $V\in\Repfd(\UqLg)$.
There exists a QSP intertwiner
\begin{align}
 \rKM{\psi,V}{z}:\shrep{V}{z}\to\shrep{\psi^*(V)}{1/z}
\end{align}
in $\End(V)(z)$.
\end{lemma}

\begin{proof}
By Theorems~\ref{thm:spectral-k} and \ref{ss:k-fd-rep-2}, for any $V\in\Repfd(\UqLg)$,
the element $\sKM{\psi}{z}$ provides an intertwiner
\begin{equation}
	\sKM{\psi,V}{z}:\Lfml{V}{z}\to\Lfml{\psi^*(V)}{1/z}\, .
\end{equation}
This is equivalent to the existence of a solution $\sKM{\psi, V}{z}\in\Lfml{\End(V)}{z}$
of a finite system of linear equations, namely
\begin{equation}\label{eq:spectral-k-intertwiner-V}
	\sKM{\psi,V}{z}\cdot\pi_{V,z}(b)=\pi_{\psi^*(V), 1/z}(b)\cdot\sKM{\psi,V}{z}\, ,
\end{equation}
where $b\in\Uqk$ runs over any finite set of generators of $\Uqk$.
Since the system is consistent and defined over $\bsF(z)$, it can be solved in $\End(V)(z)$.
\end{proof}

\subsection{Rational K-matrices}\label{ss:rational-k}
We shall now prove that, for irreducible representations,
the spectral K-matrix $\sKM{\psi,V}{z}$ is proportional to $\rKM{\psi,V}{z}
\in\End(V)(z)$, \ie it is rational up to a non-zero scalar in $\Lfml{\bsF}{z}$.

\begin{theorem}\label{thm:rational-k}
Let $V,W \in \Repfd(\UqLg)$ be irreducible representations.
\begin{enumerate}\itemsep0.25cm
\item
The space of QSP intertwiners $\Lfml{V}{z}\to\Lfml{\psi^*(V)}{1/z}$ is one-dimensional.
\item
There exists a formal Laurent series $g_{V}(z)\in\Lfml{\bsF}{z}$ and a non-vanishing
rational operator $\rKM{\psi,V}{z}\in\End(V)(z)$ such that
\begin{align}\label{eq:rational-decomp-k}
	\sKM{\psi,V}{z}= g_{V}(z)\cdot\rKM{\psi,V}{z}\, .
\end{align}
\item
The operators $\rKM{\psi,V}{z}$ and $\rKM{\psi,W}{w}$ satisfy the generalized reflection equation in $\End(V\ten W)(z,w)$
\begin{align}\label{eq:rational-tw-RE}
		\rRM{\psi^*(W)\, \psi^*(V)}{\tfrac{w}{z}}_{21}&\cdot 1\ten\rKM{\psi, W}{w}
		\cdot\rRM{\psi^*(V)W}{zw}\cdot\rKM{\psi, V}{z}\ten 1=\\
		&=\rKM{\psi, V}{z}\ten 1\cdot\rRM{\psi^*(W)\, V}{zw}_{21}\cdot 1\ten\rKM{\psi,W}{w}
		\cdot\rRM{VW}{\tfrac{w}{z}}\, ,
\end{align}
where $\rRM{VW}{z}$ is the rational R-matrix (see \ref{ss:rational-R}), and
\eq{
\rRM{WV}{z}_{21}\coloneqq(1\, 2)\circ\rRM{WV}{z}\circ (1\, 2).
}
\end{enumerate}
\end{theorem}

From Remark~\ref{rmk:spectral-k} (2), we get the following

\begin{corollary}
	Assume $\shift{z}(\gau)=\gau$.
	Let $V\in\Repfd(\UqLg)$ be irreducible.
	The universal K-matrix descends to a formal series operator
	$\sKM{\psi, V}{z}\in\fml{\End(V)}{z}$, endowed with a factorization
	\begin{align}
		\sKM{\psi,V}{z}=g_V(z)\cdot\rKM{\psi,V}{z}
	\end{align}
	where $g_V(z)\in\fml{\bsF}{z}$ and the entries of $\rKM{\psi,V}{z}\in\End(V)(z)$
	are rational functions over $\bsF$ regular at $z=0$.
\end{corollary}

The proof of Theorem~\ref{thm:rational-k} is carried out in Sections~\ref{ss:proof-rational-1}-\ref{ss:proof-rational-2}

\subsection{Proof of Theorem~\ref{thm:rational-k}: part (1)}\label{ss:proof-rational-1}
We say that an irreducible $V\in\Repfd{\UqLg}$ is \emph{generically QSP irreducible}
if $\Lfml{V}{z}$ is irreducible as a representation over $\Lfml{\Uqk}{z}$.
Such a condition is the natural counterpart of the generic irreducibility of the tensor product $V\ten \Lfml{W}{z}$ in Theorem \ref{thm:rational-R}, which holds for any pair of irreducible representations $V,W$.\\

The condition of generic irreducibility clearly depends upon the choice of a grading shift.
In Corollary~\ref{cor:QSP:irreducible}, we prove that every irreducible $\UqLg$-representation is
generically QSP irreducible with respect to the {\em principal} grading shift.
It is then clear that, in this case, (1) follows by Schur's lemma.
Namely, let $\cK_1,\cK_2\in\Lfml{\End(V)}{z}$ be two  solutions of \eqref{eq:spectral-k-intertwiner-V}.
Let
\eq{
	\Pfml{\bsF}{z}\coloneqq \bigcup_{n>0}\Lfml{\bsF}{z^{1/n}}
}
be the field of Puiseux series over $\bsF$, i.e. the algebraic closure of $\Lfml{\bsF}{z}$, and set
$\Pshrep{V}{z}\coloneqq V\ten\Pfml{\bsF}{z}$.
The composition $\cK_2^{-1}\cK_1:\Pfml{V}{z}\to\Pfml{V}{z}$ is an intertwiner.
Therefore, by QSP irreducibility and Schur's lemma, there exists $g(z)\in\Pfml{\bsF}{z}$ such that $\cK_1=g(z)\cK_2$.
Clearly, since both operators are defined over $\Lfml{\bsF}{z}$, one has $g(z)\in\Lfml{\bsF}{z}$.\\

The irreducibility result does not immediately carry over to the case of the $\tau$-minimal grading shift.
Instead, we generalize (1) by proving that the above result on the one-dimensionality of the space of QSP intertwiners \eqref{eq:spectral-k-intertwiner-V} for the principal grading implies the one-dimensionality of the space of QSP intertwiners for the $\tau$-minimal grading shift.\\

Let ${\sf pr}: \aQ \to \Z$ be the group homomorphism defined by $\al_i \mapsto 1$ for all $i \in \wh I$ so that, in the notation of Section \ref{ss:tau-grading}, $\Sigma_z^{\sf pr}$ denotes the principal grading shift.
Consider any extension of ${\sf pr}|_{\sfQ}$ and $s_\tau|_{\sfQ}$ to group homomorphisms from $\sfP\to\Q$, also denoted ${\sf pr}$ and $s_\tau$, respectively.
Note that the extended ${\sf pr}$ and $s_\tau$ will in fact take images in $\frac{1}{m} \Z$ for some positive integer $m$.\\

Note that in the following we shall regard $V$ as a $\sfP$-graded vector space.
Let $M_{V}(z)$ be the linear operator on $V(z^{1/m}) \subset V\{z\}$ given by
$M_{V}(z)v_{\la}=z^{\sf pr(\la)}v_{\la}$ for any weight vector $v_\la$ ($\la \in\sfP$).
Let $h$ be the Coxeter number of $\fkg$ and set $h_\tau \in \tfrac{1}{2}\Z$ by
\eq{
	h_\tau := \begin{cases} h & \text{if } \tau(0)=0, \\ \frac{h+1}{2} & \text{if } \tau(0) \ne 0. \end{cases}
}
Since $h=\operatorname{ht}(\drv{})$, it follows that $M_V(z)$ intertwines between the
{\em principal} shifted action and the {\em $\tau$-minimal }shifted action, \ie
$\Ad(M_V(z)) \circ \pi^{s_{\tau}}_{z^{h_\tau},V} = \pi^{\sf pr}_{z,V}$.
Finally, we observe that the linear map
\eq{
	K_{\psi,V}^{\sf pr}(z) \mapsto
	M_{\psi^*(V)}(z^{-1})^{-1} \cdot K_{\psi,V}^{\sf pr}{z} \cdot M_V(z)\eqqcolon K_{\psi,V}^{s_{\tau}}(z)
}
defines a bijection between the spaces of principal and $\tau$-minimal QSP intertwiners.
Since we have already shown that the space of solutions $\sKM{\zsat,V}{z}:V\{z\}\to\psi^*(V)\{1/z\}$ of the linear system
\eq{ \label{kV:intw:principal}
	\sKM{\psi,V}{z}\cdot\pi^{\sf pr}_{V,z}(b) = \pi^{\sf pr}_{\psi^*(V), 1/z}(b)\cdot\sKM{\psi,V}{z}\,\qq \text{for all } b \in\Uqk
}
is one-dimensional, the result follows.

\subsection{Proof of Theorem~\ref{thm:rational-k}: part (2) and (3)}\label{ss:proof-rational-2}

Part (2) follows immediately from part (1) and Lemma \ref{lem:rational-intertwiner}.
Namely, let $\rKM{\psi}{z}\in\End(V)(z)$ be any rational solution of \eqref{eq:spectral-k-intertwiner-V}, whose existence is guaranteed by Lemma~\ref{lem:rational-intertwiner}. By (1), there exists $g_V(z)\in\Lfml{\bsF}{z}$ such that
the identity $\sKM{\psi,V}{z}= g_{V}(z)\cdot\rKM{\psi,V}{z}$ holds.\\

It remains to prove part (3).
Let $\rRM{VW}{z}=f_{VW}(z)^{-1}\sRM{VW}{z}$ be the rational R-matrix (cf.~Thm.~\ref{thm:rational-R}).
Then, (2) reduces to prove that
\begin{align}\label{eq:scalar-identity}
	f_{\psi^*(W)\, \psi^*(V)}(\tfrac{w}{z})\cdot g_{V}(z)&\cdot f_{\psi^*(V)\, W}(zw)\cdot g_{W}(w)=\\
	=&g_{V}(z)\cdot f_{\psi^*(W)\, V}(zw)\cdot g_{W}(w) \cdot f_{VW}(\tfrac{w}{z})
\end{align}
 Note that,
by \cite[Thm. 8.10]{appel-vlaar-20}, we have $\RM{21}^{\psi\psi}=\RM{\psi}\RM{}\RM{\psi}^{-1}$.
Therefore,
\begin{align}
\sRM{\psi^*(W)\, \psi^*(V)}{\tfrac{w}{z}}_{21}=\RM{\psi}\cdot\sRM{VW}{\tfrac{w}{z}}\cdot\RM{\psi}^{-1}=f_{VW}(\tfrac{w}{z})\rRM{\psi^*(W)\, \psi^*(V)}{\tfrac{w}{z}}_{21}\, ,
\end{align}
\ie $f_{\psi^*(W)\, \psi^*(V)}(\tfrac{w}{z})=f_{VW}(\tfrac{w}{z})$.
Similarly, replacing ${V}$ with $\psi^*(V)$, we get
\begin{align}
	f_{\psi^*(V)\, W}(zw)=f_{\psi^*(W)\, (\psi^2)^*V}(zw)=f_{\psi^*(W)\, V}(zw)\, .
\end{align}
Note that the second equality holds since $\psi^2$ is a weight zero operator.
Therefore it is constant in $z$ and does not impact the normalization function $f_{\psi^*(W)\, V}(zw)$.
The result follows.
\qedhere

\subsection{Remarks}\label{rmk:rational}

\hfill
\begin{enumerate}[leftmargin=2em]\itemsep0.25cm
\item
Our proof of the generalized reflection equation \eqref{eq:rational-tw-RE} bears some similarity in spirit with the proof of the (untwisted) reflection equation in \cite[Sec.~3]{delius-george-02} (also cf. \cite[Sec.~5.2]{delius-02} and \cite[Sec.~2.2]{delius-mackay-03}) and \cite[Prop.~6.6]{regelskis-vlaar-16} from an assumed solution of the intertwining condition \eqref{eq:spectral-k-intertwiner-V} in the special case $\psi^*(V)=V$; note that the existence of such an intertwiner follows from generic QSP irreducibility of $V$.
Both are boundary versions of the approach given in \cite{jimbo-86} for constructing solutions of the parameter-dependent Yang-Baxter equation.
It also relies on the generic QSP irreducibility of $V \ot W$ (only the case where $V$ and $W$ are evaluation representations is considered).
Furthermore, in these approaches an additional property\footnote{In \cite{delius-george-02}, this is the property that $\mathcal{R}_{VW}(z)|_{q = 1}$ is proportional to the identity.
In \cite{regelskis-vlaar-16} only the case where $V = W$ is the vector representation is studied and the additional property (known as \emph{regularity}) is that $\mathcal{R}_{VV}(1)$ is proportional to the elementwise flip of tensor factors in $V \ot V$.} of the R-matrix is needed to show that the scalar resulting from the application of Schur's lemma equals 1.

On the other hand, the universal K-matrix approach has a wider applicability: it requires only the generic QSP irreducibility of the two representations $V$ and $W$, not of $V \ot W$, and guarantees the existence of a matrix-valued rational function which is at the same time a (twisted) $U_q\fkk$-intertwiner and a solution of a (generalized) reflection equation.
\item
	The definition of the set $\GQSP$ may appear at first quite restrictive. However, for
	the proofs of Theorems~\ref{thm:spectral-k} and \ref{thm:rational-k} to work, a gauge
	transformation $\gau\in\Gg$ is required to satisfy the following three conditions:
	\vspace{0.25cm}
	\begin{enumerate}[label=(G\arabic*), start=1, leftmargin=3em]\itemsep0.25cm
		\item\label{cond:G1} $\gau\cdot\bt{\tsat}^{-1}$ is shift-invariant;
		\item\label{cond:G2} $\gau\cdot\parc{}^{-1}$ descends to an operator on shifted representations;
		\item \label{cond:G3} the automorphism $\psi^2$, where $\psi=\Ad(\gau)\circ\tsat_{q}^{-1}$, is weight preserving.
	\end{enumerate}
	\vspace{0.25cm}
	By observing the action of $\gau$ on the Cartan subalgebra at $q=1$, one checks that the
	conditions \ref{cond:G1}, \ref{cond:G2}, \ref{cond:G3} essentially determine the set $\GQSP$.
\rmkend
\end{enumerate}

 \subsection{Unitary K-matrices}\label{ss:unitarity}
 Let $V,W\in\Repfd(\UqLg)$ be irreducible representations. By Theorem~\ref{thm:rational-R},
 the rational R-matrix $\rRM{VW}{z}$ satisfies the unitarity condition
 $\rRM{VW}{z}^{-1}=(1\, 2)\circ\rRM{WV}{z^{-1}}\circ(1\, 2)$. The analogue result for
 rational K-matrices is less straightforward. Clearly, the R-matrix argument can
 be used, under suitable assumptions.

 \begin{prop}\label{prop:unitarity}
 	Suppose $V\in\Repfd{\UqLg}$ is irreducible and
 	satisfies the following properties.
 	\begin{enumerate}[label=(\alph*)]\itemsep0.25cm
 		\item $V$ is $\psi$-involutive, \ie $(\psi^2)^*(V)=V$.
 		\item There exist non-zero vectors $v,v'\in V$ and functions $f(z),f'(z)\in\Lfml{\bsF}{z}$
 		such that $\sKM{\psi, V}{z}v=f(z)v'$ and $\sKM{\psi,\psi^*(V)}{z}v'=f'(z)v$.
 	\end{enumerate}
 	Then, the following holds.
 	\begin{enumerate}\itemsep0.25cm
 		\item
 		There exists a choice
 		of rational K-matrices for $V$ and $\psi^*(V)$ such that
 		\begin{align}\label{eq:unitarity-k}
 			\rKM{\psi,V}{z}^{-1}=\rKM{\psi, \psi^*(V)}{z^{-1}}
 		\end{align}
 		\item
 		Let $\zeta\in\bsF^{\times}$. If $V(\zeta)$ is QSP irreducible, then $\rKM{\psi,V}{\zeta}$ is well--defined and invertible.
 	\end{enumerate}
 \end{prop}

 \begin{proof}
	(1) Set $\rKM{\psi, V}{z}=f(z)^{-1}\sKM{\psi,V}{z}$ and $\rKM{\psi, \psi^*(V)}{z}=f'(z)^{-1}\sKM{\psi,\psi^*(V)}{z}$.
	It is enough to observe that the composition
	\begin{equation}
		\begin{tikzcd}[column sep=1.5cm]
			V(z) \arrow[r, "\rKM{\psi, V}{z}"] &
			\psi^*(V)(z^{-1}) \arrow[r, "\rKM{\psi, \psi^*(V)}{z^{-1}}"] &
			(\psi^2)^*(V)(z)=V(z)
			{}	\end{tikzcd}
	\end{equation}
	is a $\Uqk$-intertwiner, which is the identity on $v$ and therefore
	\begin{equation}
		\rKM{\psi, \psi^*(V)}{z^{-1}}\rKM{\psi, V}{z}=\id_{V(z)}\, .
	\end{equation}
	(2) Let $m\geqslant 0$ be the order of the pole of $\rKM{\psi, V}{z}$ at $z=\zeta$. Then,
	the map
	\[
	\lim_{z\to \zeta}(z-\zeta)^m\rKM{\psi, V}{z}
	\]
	is a non-zero intertwiner $V(\zeta)\to \psi^*(V)(\zeta^{-1})$ and therefore invertible.
	However, by unitarity, $\rKM{\psi, \psi^*(V)}{z^{-1}}$ has a zero of order $m$.
	Therefore, $m=0$.
\end{proof}

 \begin{remarks}\hfill
 	\begin{enumerate}\itemsep0.25cm
 		\item
 		The condition (a) above is trivially satisfied in the semi-standard case
 		\ref{ss:distinguished-k} (2), where the twisting operator is given by the involution
 		$\chev\circ\tau$. In this case the condition $(\psi^2)^*(V)=V$ hold for {\em any}
 		$V\in\Repfd(\UqLg)$. In general, however,
 		some further adjustment is required.
 		For instance, we prove in \cite[Sec.~9]{appel-vlaar-20} in the case of $U_q\wh{\mathfrak{sl}}_2$ the twisting operator can be {\em corrected}, so as to become an involution on a given
 		irreducible evaluation representation.
 		\item
 		The unitary K-matrices induced by the condition (b) are certainly not canonical.
 		We expect however that a natural normalization does exist. In particular, we show
 		in \ref{ss:abelian-k-mx} that this problem can be partially solved for \emph{restricted
 		rank one} QSP subalgebras.
 	\end{enumerate}
 \end{remarks}

\subsection{Abelian K-matrices}\label{ss:abelian-k-mx}

Let $V\in\Repfd(\UqLg)$ be an irreducible representation. By \cite[Cor.~12.2.5]{chari-pressley}, $
V$ is generated by an $\ell$-highest weight vector (or pseudo-highest-weight vector), \ie a
weight vector $v_\lambda$ of weight $\lambda$, which is killed by $\Fg{0}$ and $\Eg{i}$,
with $i\in\fIS$, and is also a simultaneous eigenvector for all imaginary root vectors.

\begin{prop}\label{prop:res-rank-one}
	Assume $\tsat$ has restricted rank one and $\shift{z}(\gau)=\gau$.
	Let $V\in\Repfd(\UqLg)$ be a generically QSP irreducible representation with $\ell$-highest
	weight vector $v_{\lambda}\in V$. There is a unique $g_V(z)\in\fml{\bsF}{z}^{\times}$ such that
	$\sKM{\psi,V}{z}=g_V(z)\cdot\rKM{\psi,V}{z}$ and $\rKM{\psi,V}{z}\in\End(V)(z)$
	is a rational operator regular at $z=0$ and such that
	\begin{align}
		\rKM{\psi,V}{z}v_{\lambda}=\gau\cdot v_{\lambda}\, .
	\end{align}
\end{prop}

\begin{proof}
	Since $\tsat$ is of restricted rank 1, the quasi-K-matrix defined in Theorem \ref{thm:av-k-mx} satisfies $\QK{\tsat}= 1+\sum_{m \in \bbZ_{>0}}		\QK{\tsat,m\del}$ with $\QK{\tsat,m\del} \in \Uqanp_{m\del}$.
	Note that, since $\shift{z}(\gau)=\gau$, the quasi-K-matrix
	$\QK{\tsat}$ descends to the only spectral component of $\sKM{\psi,V}{z}$.
	In particular, there exists $\wt{g}_V(z)\in\fml{\bsF}{z}^{\times}$ such that
	\begin{align}\label{eq:normalization-quasi-k}
		\QK{\tsat,V}(z) v_\lambda=\wt{g}_V(z) v_\lambda\, ,
	\end{align}
	where $\QK{\tsat,V}(z)\coloneqq\pi_{V,z}(\QK{\tsat})$.
	Set $g_V(z)\coloneqq\parc{}^{-1}(\lambda)\wt{g}_V(z)$.
	It follows that $\rKM{\psi}{z}\coloneqq g_V(z)^{-1}\cdot\sKM{\psi, V}{z}$ is the unique intertwiner
	$\Lfml{V}{z}\to\Lfml{\psi^*(V)}{z^{-1}}$ such that
	\begin{align}\label{eq:normalization-k}
		\rKM{\psi,V}{z} v_\lambda=\gau\cdot v_\lambda\, .
	\end{align}
	Therefore, $\rKM{\psi,V}{z}\in\End(V)(z)$ is a rational operator.
\end{proof}

\begin{remark}
Clearly, a similar result can be achieved through normalization with respect to any vector
which is a simultaneous eigenvector for the imaginary root vectors. In particular, one
can consider any {extremal} vector (cf. \cite{kashiwara-02, chari-02}). \rmkend
\end{remark}

By combination with Proposition~\ref{prop:unitarity}, we get the following

\begin{corollary}
	Suppose that $\tsat$ has restricted rank one, $\shift{z}(\gau)=\gau$.
	Furthermore, let $V\in\Repfd(\UqLg)$ be such that $(\psi^2)^*(V)=V$.
	Let $v_{\lambda}\in V$ be a normalized $\ell$-highest weight vector such that $v_{\lambda}=v_{\lambda}$.
	Consider the rational K-matrices normalized as in Proposition~\ref{prop:res-rank-one} by the conditions
	\begin{gather}
		\rKM{\psi,V}{z}v_{\lambda}=\gau\cdot v_{\lambda}\,
		\aand
		\rKM{\psi, \psi^*(V)}{z}\gau\cdot v_{\lambda}=v_{\lambda}\, .
	\end{gather}
	Then, the unitarity condition $\rKM{\psi,V}{z}^{-1}=\rKM{\psi, \psi^*(V)}{z^{-1}}$
	holds.
\end{corollary}
\begin{proof}
It suffices to note that, owing to the definiton of $\gau\in\GQSP$, the vector $\gau\cdot v_{\lambda}\in\psi^*(V)$ is again a simultaneous eigenvector for the imaginary root vectors.
Then, the result follows by Propositions~\ref{prop:unitarity} and \ref{prop:res-rank-one}.
\end{proof}

\section{Generic restricted irreducibility}\label{s:irreducibility}

In this stand-alone section, we consider various \emph{restricted irreducibility} results involving the untwisted quantum loop algebra $U_qL\fkg$. More precisely, fix a proper subalgebra $\cA \subset U_qL\fkg$.
We are interested in the irreducible representations $V\in\Repfd(\UqLg)$ whose shifted representation $V(z)$ remains irreducible under restriction to $\cA(z)\coloneqq\cA\ten\bsF(z)$.
We prove that for a class of subalgebras, which we refer to as \emph{modified nilpotent},
every irreducible representation over $\UqLg$ remains generically irreducible, with respect to the principal grading shift, under
restriction to $\cA$ (cf.~ Theorem \ref{thm:deformedN:irreducible}). As a corollary, we obtain
generic QSP irreducibility
(cf.~Corollary~\ref{cor:QSP:irreducible}).

\subsection{Notations}
Let $U_qL\fkg$ be an untwisted quantum loop algebra.
We use the shorthand notations $\wh U^+$, $\wh U^0$ and $\wh U^-$ to denote the subalgebras of $U_qL\fkg$ generated by all $E_i$, all $K_i^{\pm 1}$, and all $F_i$, respectively.
We will also consider the subalgebra $\wh U^{\ge 0} := \wh U^+ \cdot \wh U^0$.
In this section, for any $V\in\Repfd(\UqLg)$, we denote by $V(z)$ the shifted representation defined by the principal grading shift $\Sigma^{\sf pr}_z$  (cf.~Section~\ref{ss:tau-grading}).
Recall that $\Sigma^{\sf pr}_z$ is the morphism of $U_qL\fkg[z,z^{-1}]$ defined by
\eq{
\shift{z}^{\sf pr}(K_i^{\pm 1}) = K_i^{\pm 1}, \qq \shift{z}^{\sf pr}(E_i) = z E_i, \qq \shift{z}^{\sf pr}(F_i) = z^{-1} F_i \qq \text{for all } i \in \wh I.
}

\subsection{Principal irreducibility for modified nilpotent subalgebras}

We call a subalgebra $\cA \subseteq U_qL\fkg$ a \emph{modified nilpotent subalgebra} if it is generated by elements of the form $\wt F_i \coloneqq F_i + E'_i$ for $i \in \wh I$, where $E'_i \in \wh U^{\ge 0}$.

\begin{theorem} \label{thm:deformedN:irreducible}
	Let $\cA$ be a modified nilpotent subalgebra.
	Let $V\in\Repfd(\UqLg)$ be irreducible.
	Then, $V(z)$ remains irreducible under restriction to $\cA(z)$.
\end{theorem}

We will give the proof of this theorem in Section \ref{sec:gnrc:irred:modified}.

Let $\theta$ be a pseudo-involution of $\wh \fkg$ of the second kind with associated generalized Satake diagram $(X,\tau)$.
Also let $(\bm \ga,\bm \si) \in {\bm \Gamma}\times {\bm \Sigma}$ and consider the case
\eq{
	E'_i = \begin{cases}
		\parc{i} \theta_q(F_i) + \pars{i} K_i^{-1} & \text{if } i \in \wh I \backslash X, \\
		0 & \text{if } i \in X.
	\end{cases}
}
Since $\theta_q(F_i) \in \wh U^+ \cdot K_i^{-1}$ if $i \in \wh I \backslash X$, we clearly have $E'_i \in \wh U^{\ge 0}$.
Hence, $\wt F_i$ specializes to the generator $B_i$ of $U_q\fkk$ defined by \eqref{Bi:def} (recall that we may view $U_q\fkk$ as a coideal subalgebra of the quantum loop algebra $U_qL\fkg$).
In other words, QSP subalgebras contain modified nilpotent subalgebras of $U_qL\fkg$.
Hence Theorem \ref{thm:deformedN:irreducible} specializes to the following statement of generic QSP irreducibility.

\begin{corollary} \label{cor:QSP:irreducible}
	Let $V\in\Repfd(\UqLg)$ be irreducible.
	Let $\theta$ be a pseudo-involution of $\wh \fkg$ of the second kind and
	$(\bm \ga,\bm \si) \in{ \bm\Gamma} \times {\bm\Sigma}$.
	Set
	\eq{
		U_q\fkk^- := \langle B_{i;\theta,\ga_i,\si_i} \rangle_{i \in \wh I} \subseteq U_q\fkk, \qq U_q\fkk^-(z) := U_q\fkk^- \ot \bbF(z).
	}
	Then $V(z)$ is irreducible as a representation over $U_q\fkk^-(z)$
	(and hence over $U_q\fkk(z)$).
\end{corollary}

\subsection{Restricted irreducibility of finite-dimensional representations with respect to nilpotent subalgebras}

In this subsection we will prove the following stepping stone for Theorem \ref{thm:deformedN:irreducible}: irreducible finite-dimensional representations of $U_qL\fkg$ are also irreducible as $\wh U^+$-representations\footnote{
It contrasts nicely with the situation for the quantum group $U_q\fkg$ of finite type: its nontrivial finite-dimensional irreducible representations do not remain irreducible when restricted to $U_q\fkn^+$.
For such representations there is an essentially unique $U_q\fkn^+$-cyclic vector, which however is not $U_q\fkn^+$-cocyclic.
}.
To our best knowledge, this result is not stated in the literature, although the analogous result for $\wh U^{\ge 0}$ is known, see \cite{bowman-07,hernandez-jimbo-12}, and will be the starting point for our proof of the stronger result involving $\wh U^+$.

First we recall some basic theory of irreducible finite-dimensional representations of $U_qL\fkg$.
Let $V$ be an irreducible finite-dimensional representation of $U_qL\fkg$.
Note that $V$ is also a finite-dimensional representation of the quantum group $U_q\fkg$ of finite type (its irreducible summands are necessarily connected by the action of $E_0$ and $F_0$).
Hence we obtain a weight decomposition $V = \bigoplus_{\la \in P} V_\la$ where
\eq{
V_\la = \{ v \in V \, | \, \forall i \in I \, K_i \cdot v = q_i^{\la(h_i)} v \}.
}
Note that $K_0$ acts on each $V_\la$ as multiplication by $q_i^{\la(h_0)} = q^{-(\la,\hrt)}$.
Consider the unique $\Z$-linear projection $\ol{\cdot}: \wh Q \to Q$ such that $\ol{\al_i}=\al_i$ ($i \in I$) and $\ol{\del}=0$.
For all $i \in \wh I$ and all $\la \in P$, we have
\eq{
E_i \cdot V_\la \subseteq V_{\la + \ol{\al_i}}, \qq F_i \cdot V_\la \subseteq V_{\la - \ol{\al_i}}.
}
Finally, the \emph{support} of $V$ is the finite set
\eq{
\mathsf{Supp}(V) := \{ \lambda \in P \, | \, V_\lambda \ne \{0\} \}.
}
As foreshadowed, we start with the following restricted irreducibility result.

\begin{prop}[{\cite[Prop.~3.5]{hernandez-jimbo-12} or \cite[Thm. 2.3 (ii) for $\eps = (1)^n$]{bowman-07}}] \label{prop:N:irreducible:prep}
Every finite-dimensional irreducible representation of $\UqLg$ remains irreducible under restriction to $\wh U^{\ge 0}$.
\end{prop}

Fix an irreducible representation $\pi_V: \UqLg \to \End(V)$ where $V$ is finite-dimensional.
From Proposition \ref{prop:N:irreducible:prep} we deduce some useful technical results, the first of which directly generalizes \cite[Cor.~2.7]{chari-greenstein-05}.

\begin{lemma} \label{lem:N:irreducible:prep}
Let $\la \in \Supp(V)$ and $v_\la \in V_\la \backslash \{0\}$ be arbitrary.
The following statements are satisfied.
\begin{enumerate}
\item We have $V = \wh U^+ \cdot v_\la$.
\item For all $\mu \in \Supp(V)$ there exist $\ell \in \Z_{\ge 0}$ and $\bm i \in \wh I^\ell$ with the property that $E_{i_1} \cdots E_{i_\ell} \cdot v_\la \in V_\mu \backslash \{0\}$.
\end{enumerate}
\end{lemma}

\begin{proof}\mbox{}
\begin{enumerate}
\item
This follows from Proposition \ref{prop:N:irreducible:prep}, the decomposition $\wh U^{\ge 0} = \wh U^+ \cdot \wh U^0$ and the fact that $\wh U^0 \cdot v_\la = \bbF v_\la$.
\item
Note part (1) yields the existence of $x \in \wh U^+$ such that $x \cdot v_\la \in V_\mu \backslash \{0\}$.
Each monomial in the $E_i$ ($i \in \wh I$) occurring in $x$ must send $v_\la$ into a weight space.
Hence without loss of generality we may assume that all monomials occurring in $x$ map $v$ to $V_\mu$.
Since there must at least be one such monomial which does not annihilate $v_\la$, we obtain the result. \hfill \qedhere
\end{enumerate}
\end{proof}

We provide two further definitions.
For all $v \in V$ there exists a unique subset $\mathsf{Supp}(v) \subseteq \mathsf{Supp}(V)$ such that
\eq{
v = \sum_{\la \in \mathsf{Supp}(v)} v_\la, \qq v_\la \in V_\la \backslash \{0\}.
}
Note that $|\mathsf{Supp}(v)|=1$ if and only if $v$ is a weight vector.
For all $i \in \wh I$ we obtain
\eq{ \label{support:shift}
\mathsf{Supp}(E_i \cdot v) \subseteq {\sf Supp}(v) + \ol{\al_i}, \qq  \mathsf{Supp}(F_i \cdot v) \subseteq \mathsf{Supp}(v) - \ol{\al_i},
}
and hence
\eq{ \label{width:bound}
|\mathsf{Supp}(E_i \cdot v)|,|\mathsf{Supp}(F_i \cdot v)| \le |\mathsf{Supp}(v)|.
}

Let $\la \in \Supp(V)$ and $v_\la \in V_\la \backslash \{0\}$.
By Lemma \ref{lem:N:irreducible:prep} (2), the following definition makes sense:
\eq{ \label{depth:def1}
d^+(v_\la) := \min\{ \ell \in \Z_{\ge 0} \, | \, \exists \bm i \in \wh I^\ell \, E_{i_1} \cdots E_{i_\ell} \cdot v_\la \in V_{\la_0} \backslash \{0 \} \} \in \Z_{\ge 0}.
}
More generally, for any $v \in V \backslash \{0\}$ we define
\eq{
d^+(v) := \min \{ d^+(v_\la) \, | \, \la \in \mathsf{Supp}(v) \} \in \Z_{\ge 0}.
}
Note that $d^+(v) = 0$ if and only if $\la_0 \in {\sf Supp}(v)$.
We have the following useful result.

\begin{lemma} \label{lem:N:irreducible:prep2}
Let $v \in V \backslash \{0\}$.
If $d^+(v)>0$ then there exists $i \in \wh I$ such that $E_i \cdot v \ne 0$ and $d^+(E_i \cdot v)<d^+(v)$.
\end{lemma}

\begin{proof}
Set $\ell := d^+(v) \in \Z_{>0}$.
The definition of $d^+$ implies that $\la_0 \notin {\sf Supp}(v)$ and there exists $\la \in {\sf Supp}(v)$ such that $d^+(v_\la) = \ell$.
By Lemma \ref{lem:N:irreducible:prep} (2) there exists $\bm i \in \wh I^\ell$ such that $E_{i_1} \cdots E_{i_\ell} \cdot v_\la \in V_{\la_0} \backslash \{0 \}$.
Set $i=i_\ell$.
Then $d^+(E_i \cdot v_\la) = \ell-1$.
The relation \eqref{support:shift} and the definition of $d^+$ now imply $d^+(E_i \cdot v) < \ell$.
\end{proof}

Now we are ready to state and prove the first main result of this section.

\begin{theorem} \label{thm:N:irreducible}
Every finite-dimensional irreducible representation of $\UqLg$ remains irreducible under restriction to $\wh U^+$.
\end{theorem}

\begin{proof}
We will show that $\wh U^+ \cdot v = V$ for all $v \in V \backslash \{0\}$.
Considering Lemma \ref{lem:N:irreducible:prep} (1), it remains to show that $\wh U^+ \cdot v$ contains a weight vector.
We will prove this by induction with respect to $s := |\Supp(v)| \in \Z_{>0}$.
This is immediate for $s=1$.
It suffices to prove that the statement is true for a given $s \in \Z_{>1}$, assume that the statement is true for all $v \in V$ such that $|\Supp(v)|<s$.\\

Suppose on the contrary that there exists $v \in V$ such that $ |\mathsf{Supp}(v)|=s$ and $\wh U^+ \cdot v$ does not contain a weight vector; among all those $v$, choose one with a minimal value of $d^+(v)$.
We shall deduce a contradiction and hence conclude that such $v$ does not exist.
We first derive inequalities using some casework.
\begin{enumerate}
\item
Suppose $d^+(v)=0$.
Then $\la_0 \in {\sf Supp}(v)$.
Since $|{\sf Supp}(v)|>1$, we may choose $\mu \in {\sf Supp}(v) \backslash \{ \la_0 \}$.
By Lemma \ref{lem:N:irreducible:prep} (2) there exists $\bm i \in \wh I^\ell$ with $\ell = d^+(\mu)$ such that $E_{i_1} \cdots E_{i_\ell} \cdot v_\mu \in V_{\la_0} \backslash \{0\}$.
We must have $\mu + \ol{\al_{i_1} + \ldots + \al_{i_\ell}} = \la_0$ with $\ol{\al_{i_1} + \ldots + \al_{i_\ell}} \ne 0$.
From ${\sf Supp}(V) \subset \la_0 - Q^+$ we obtain $\ol{\al_{i_1} + \ldots + \al_{i_\ell}} \in Q^+$.
Since $\la_0 + \ol{\al_{i_1} + \ldots + \al_{i_\ell}} \in \la_0 + Q^+$, using ${\sf Supp}(V) \subset \la_0 - Q^+$ again we obtain $\la_0 + \ol{\al_{i_1} + \ldots + \al_{i_\ell}} \notin {\sf Supp}(V)$.
Hence $E_{i_1} \cdots E_{i_\ell} \cdot v_{\la_0} = \{0 \}$.
As a consequence, $|{\sf Supp}(E_{i_1} \cdots E_{i_\ell} \cdot v)| < |{\sf Supp}(v)|$.
\item
Suppose $d^+(v)>0$.
By Lemma \ref{lem:N:irreducible:prep2} there exists $i \in \wh I$ such that $E_i \cdot v \ne 0$ and $d^+(E_i \cdot v) < d^+(v)$.
From the minimality of $v$ and \eqref{width:bound} we deduce that $|{\sf Supp}(E_i \cdot v)| < |{\sf Supp}(v)|$.
\end{enumerate}
In either case, we obtain an inequality of the form $|{\sf Supp}(E_{i_1} \cdots E_{i_\ell} \cdot v)|<|{\sf Supp}(v)|$.
But now the inclusion $\wh U^+ \cdot (E_{i_1} \cdots E_{i_\ell} \cdot v) \subseteq \wh U^+ \cdot v$ furnishes a contradiction with the induction hypothesis.
This completes the proof.
\end{proof}

By pulling back $\pi_V$ by $\om$, we immediately obtain the following result, which is more useful for our purposes.

\begin{corollary} \label{crl:N:irreducible}
Every finite-dimensional irreducible representation of $\UqLg$ remains irreducible under restriction to $\wh U^-$.
\end{corollary}

\subsection{Proof of Theorem \ref{thm:deformedN:irreducible}} \label{sec:gnrc:irred:modified}

The proof of the main theorem of this section relies on the following basic linear-algebraic result.

\begin{lemma} \label{lem:linearalgebra}
Let $z$ be an indeterminate and $V$ a finite-dimensional $\mathbb{F}$-linear space.
Let $\mathcal M(z) := \{ M_i(z) \}_{i \in \wh I}$ be a finite set of endomorphisms of $V(z) := V \ot \mathbb{F}(z)$ depending polynomially on $z$.
We may view $\mathcal M := \{ M_i(0) \}_{i \in \wh I}$ as a set of endomorphisms of $V$.
Suppose $V$ has no nontrivial proper $\mathcal M$-invariant subspace.
Then $V(z)$ has no nontrivial proper $\mathcal M(z)$-invariant subspace.
\end{lemma}

\begin{proof}
Note that evaluation at 0 defines a linear map ${\sf ev}_0$ from $V[z]:=V \ot {\mathbb F}[z] \subset V(z)$ to $V$.
Let $S(z)$ be a nontrivial $\mathcal M(z)$-invariant subspace of $V(z)$ and abbreviate $S[z] := S(z) \cap V[z]$.
As a consequence, for all $s(z) \in S[z]$ and for all $i \in \wh I$ we have
\eq{
M_i(z) \cdot s(z) \in S[z].
}
It follows that ${\sf ev}_0(S[z])$ is a nontrivial $\mathcal M$-invariant subspace of $V$ and therefore ${\sf ev}_0(S[z])=V$.
Extending the scalars to $\mathbb{F}(z)$, we obtain $S(z) = V(z)$, as required.
\end{proof}

We can now complete the proof of Theorem \ref{thm:deformedN:irreducible}.
Since $\wt F_i - F_i \in \wh U^{\ge 0}$ for all $i \in \wh I$, the action of $z \wt F_i \in U_qL\fkg(z)$ on $V(z)$ is polynomial in $z$.
We set $M_i(z)$ equal to this action and note that $M_i(0)$ is equal to the action of $zF_i$, which descends to an action on $V$.
Combining Corollary \ref{crl:N:irreducible} and Lemma \ref{lem:linearalgebra} we deduce that $V(z)$ has no nontrivial proper $\{ M_i(z) \}_{i \in \wh I}$-invariant subspace.
Since $z$ is invertible in $\bbF(z)$, we may replace $z\wt F_i$ by $\wt F_i$ and we obtain the desired result.

\section{Solutions of the reflection equations}\label{s:recover-RE}
In this section we connect to the existing theory of solutions of the parameter-dependent reflection equation as initiated in the fundamental papers \cite{cherednik-84,sklyanin-88}.

\subsection{The standard reflection equations}
Recall that, by Theorem \ref{thm:rational-R}, for any irreducible $V,W\in\Repfd{\UqLg}$,
there exists a rational R-matrix $\rRM{VW}{z}$. Thus, on the tensor product $\shrep{V}{z}\ten\shrep{W}{w}$ we consider the (standard) reflection equation
\eq{ \label{eq:rational-original-RE}
	\begin{aligned}
		\rRM{WV}{\tfrac{w}{z}}_{21} & \cdot \id \ten\rKM{W}{w} \cdot \rRM{VW}{zw} \cdot \rKM{V}{z} \ten \id =\\
		&=\rKM{V}{z} \ten \id \cdot \rRM{WV}{zw}_{21} \cdot \id \ten \rKM{W}{w} \cdot \rRM{VW}{\tfrac{w}{z}}
	\end{aligned}
}
for unknown operators $\rKM{V}{z}$ and $\rKM{W}{w}$. These are typically assumed to be invertible matrix-valued formal Laurent series (or generically invertible matrix-valued meromorphic functions when $q\in\bbC$). There is a rich literature of work comprehensively cataloguing
the solutions of the standard reflection equation, see \eg \cite{malara-lima-santos-06,regelskis-vlaar-16,regelskis-vlaar-18}
and references therein.\\

The interplay between the reflection equation and QSP subalgebras has been deeply studied and explored, see e.g. \cite{delius-02,delius-george-02,delius-mackay-03,baseilhac-belliard-10,regelskis-vlaar-16,baseilhac-tsuboi-18,vlaar-weston-20}.
In this context, it is natural to ask whether the equation \eqref{eq:rational-original-RE}
can be obtained as a special case of the generalized reflection equation \eqref{eq:rational-tw-RE} with respect to a twisting operator $\psi$ which leaves the representations $V$ and $W$
invariant under pullback. In the next section we prove that this is indeed the case.

\begin{remark}
There exists a common variant of \eqref{eq:rational-original-RE}, which is related to the dual representations  \cite{kulish-sklyanin-1992,kulish-et-al-1993,bowcock-et-al-1995,
bowcock-et-al-1996,doikou-2000}, \ie
\eq{
	\label{eq:rational-transposed-RE}
	\begin{aligned}
		\rRM{WV}{\tfrac{w}{z}}_{21}^{{\sf t}_V {\sf t}_W} & \cdot \Id_V \ot \trKM{W}{w} \cdot ( \rRM{VW}{zw}^{-1})^{\sf t_V} \cdot \trKM{V}{z} \ot \Id_W = \\*
		&\qq =  \trKM{V}{z} \ot \Id_W \cdot (\rRM{WV}{zw}^{-1}_{21})^{\sf t_W} \cdot \Id_V \ot \trKM{W}{w}\cdot\rRM{VW}{\tfrac{w}{z}}
	\end{aligned}
}
where ${\sf t}_V$ and ${\sf t}_W$ denote the transposition on the first and second component,
respectively. The equation \eqref{eq:rational-transposed-RE} is associated to the case where
the pullback through the twisting operator $\psi$ gives back the dual representations of $V$ and
$W$. However, for brevity, we shall not discuss this case here.
\rmkend
\end{remark}

\subsection{Solutions through QSP intertwiners}\label{ss:our-solutions}
We claim that the rational K-matrices constructed in Theorem~\ref{thm:rational-k} do satisfy the reflection equation \eqref{eq:rational-original-RE}, for a suitable choice of the twisting operator $\psi$ and eventually of the QSP subalgebra itself.\\

Let $(X,\tau)\in\gsat{\aff{A}}$ with pseudo-involution $\tsat$, $(\parc{},\pars{})\in\Parsetc\times\Parsets$ such that $\parc{}(\drv{})=1$, and $\Uqk\subset\Uqagp$ the corresponding QSP subalgebra.
We consider an auxiliary generalized Satake diagram $(Y_0,\eta_0)\in\gsat{\aff{A}}$ of restricted rank one and defined as follows.
Set
\eq{
Y_0\coloneqq\aIS\setminus\{0,\tau(0)\}, \qq \eta_0(0)=\tau(0), \qq \eta_0\vert_{Y_0}=\oi_{Y_0}.
}
Let $\zsat_0$ be the corresponding pseudo-involution.
Note that, by considering the gauge transformation
$\gau_0\coloneqq\bt{\zsat_0}^{-1}\bt{\tsat}\in\GQSP$, we obtain the QSP admissible
twisting operator $\psi_0=\zeta_{0,q}^{-1}\circ\eta_0\circ\tau$. As we pointed out in
Remark~\ref{rmk:distinguished} (4), $\eta_0$ and $\tau$ commute and the twisting operator
acts as the involution $\eta_0\circ \tau$ on a large class of representations.

\begin{theorem}
Let $\Uqk\subset\UqLg$ be a {\em restrictable} affine QSP subalgebra, \ie $0\not\in X$ and $\tau(0)=0$. Let $V, W\in\Repfd{\UqLg}$ be irreducible representations, which remain irreducible under restriction to $\Uqg$.
\begin{enumerate}\itemsep0.25cm
\item The rational K-matrix $\rKM{V}{z}\coloneqq\rKM{\psi_0,V}{z}$ is
a QSP intertwiner
\begin{equation}
\rKM{V}{z}:\shrep{V}{z}\to\shrep{V^{\eta_0\tau}}{1/z}\,,
\end{equation}
where $V^{\eta_0\tau}=(\eta_0\tau)^*(V)$ (and similarly for $\rKM{W}{z}\coloneqq\rKM{\psi_0,W}{z}$).
\item The K-matrices $\rKM{V}{z}$ and $\rKM{W}{w}$ satisfy the {\em diagrammatic}
reflection equation
\begin{align}\label{eq:bala-kolb-re}
	\rRM{WV}{\tfrac{w}{z}}_{21}&\cdot \id\ten\rKM{W}{w}
	\cdot{\bf R}_{V\,W}^{\eta_0\tau}(zw)\cdot\rKM{V}{z}\ten \id=\\
	&=\rKM{V}{z}\ten \id\cdot{\bf R}_{W\,V}^{\eta_0\tau}(zw)_{21}\cdot \id\ten\rKM{W}{w}
	\cdot\rRM{VW}{\tfrac{w}{z}}\, .
\end{align}
\item Whenever $\tau=\eta_0$, the K-matrices $\rKM{V}{z}$ and $\rKM{W}{w}$ satisfy the
standard reflection equation \eqref{eq:rational-original-RE}.
\end{enumerate}
\end{theorem}

\begin{proof}
The result follows immediately from Theorem~\ref{thm:rational-k}.
Namely, it is enough to observe that, if $V\in\Repfd{\UqLg}$ is an irreducible representation which remains irreducible under restriction to $\Uqg$, then the defining data of $V$ as a representation over $\UqLg$ can be expressed entirely in terms of $\Uqg$.
Moreover, when $\Uqk$ is a restrictable QSP subalgebra, then $Y_0=I$ and $\zsat_0$ is the maximal pseudo-involution with respect to the finite Dynkin diagram.
Therefore, $\zsat_{0,q}$ is the identity on $\Uqg$, so that $\psi_0^*(V)=(\eta_0\tau)^*(V)$.
The result follows.
\end{proof}

\begin{remarks}\hfill
	\begin{enumerate}\itemsep0.25cm
		\item In the case of $U_qL\mathfrak{sl}_N$, every evaluation representation
		satisfy the conditions of the Theorem.
		\item In particular, the result applies to the case of the generalized
		$q$-Onsager algebras, see \cite{baseilhac-belliard-10}, with respect to any evaluation representation of $U_qL\mathfrak{sl}_N$.
		\item The constant (\ie parameter-independent) version of the diagrammatic reflection
		equation appeared first in the work of Balagovic-Kolb on the universal K-matrices for finite-type
		quantum groups \cite{balagovic-kolb-19}.
		\item The constraint $0 \notin X$ is not strictly necessary. However, when $\Uqk$ is
		restrictable, the K-matrices are regular at $z=0$ and one can show that,
		as in the case of the R-matrix, $\rKM{V}{0}$ reduces to the K-matrix of the underlying finite-type QSP subalgebra.
	\end{enumerate}
\end{remarks}

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